problem solving of universal set

Universal Set in Math – Definition, Symbol, Examples, Facts, FAQs

What is universal set in math, venn diagram of universal set, difference between universal set and union of sets, solved examples on universal set, practice problems on universal set, frequently asked questions on universal set.

The universal set (symbol: U ) is a set that contains all the elements of other related sets with respect to a given subject. It is a larger set that contains elements of all the related sets, without any repetition.

In mathematics, a set is defined as a collection of distinct, well-defined objects. 

Examples: the set of whole numbers, the set of months in a year, the set of positive even integers, etc. 

The universal set, as the term “universal” suggests, is the set containing all the relevant elements for the subject under discussion. It is the superset of all basic sets related to the given topic. 

Example: The set of real numbers is the universal set for the set of integers , the set of rational numbers , the set of natural numbers, and the set of whole numbers .

Consider an example in day-to-day life. If we are listing or studying animals, the universal set would be the set of all the animals in the world. The set of all animals in each country can be considered as a subset of this universal set.

Universal Set: Definition

The collection or set of all the elements in regard to a particular subject under discussion is known as a universal set.

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Universal Set: Symbol

The capital letter “U” is usually used to denote a universal set. However, it is not a standard notation. The symbol “Ω” is also used to denote a Universal set. You can choose a symbol based on the topic at hand. 

A Venn diagram is an illustration or pictorial representation that uses circles to show the relationships among things. Circles that overlap or intersect show that there are common elements present while circles that do not overlap do not share any common traits.

Venn diagrams were created by mathematician John Venn and are widely used today in computer science, problem-solving, and set theory.

When drawing Venn diagrams, we use a rectangular box to represent the universal set. Circles are used to represent sets. All the subsets are drawn inside this box.

Consider the following example: 

Set $A = \left\{2, 4, 6, 7, 9\right\}$

Set $B = \left\{2, 3, 4, 5\right\}$

Universal Set $U = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$

The universal set and the union of two or more sets can be confusing, but both these terms are different.

Universal Set

We know that a universal set is a set that contains all the elements of other sets and is represented by the symbol “U.”  

Union of Two Sets

The union of two sets A and B is the set that contains elements that are either in A or in B or in both A and B. The union of two sets A and B is a new set represented by A ∪ B. 

In the Venn diagram shown below, the yellow shaded region represents the union of sets A and B. The entire rectangle with a purple border represents the universal set. It is represented by the letter “U” at the top right corner.

Example: Let’s consider three sets.

Set $A =$ Set of positive integers $= \left\{1, 2, 3, …\right\}$ 

Set $B =$ Set of negative integers $= \left\{\;-\;1, -2, \;-\;3, …\right\}$

Set $C = \left\{0\right\}$

The universal set for the sets A, B, and C is the set of integers given by

$U = \left\{…, \;-\;3, \;-\;2, \;-\;1, 0, 1, 2, 3, …\right\}$

The union of sets A and B is given by

$A \cup B = \left\{…, \;-\;3, \;-\;2, \;-\;1, 1, 2, 3, …\right\}$

Note that 0 is not present in the union of the sets A and B. However, it is present in the universal set.

Complement of Universal Set

Every set has a complement. The complement of a set contains all the elements from the universal set that are not present in the given set. Thus, if an element is present in a complement of a set, it is not present in the set.

The complement of the universal set is the empty set $\left(\text{symbol:}\; ∅\; \text{or} \left\{\right\}\right)$

Universal Set and Its Subsets

As the universal set contains all the elements of other sets, all related sets are subsets of the universal set. 

If every element in a set A is also an element in a set B, then A is a subset of B. The subset is represented by the symbol $\subseteq$ . Note that the symbol $\subset$ is used to represent a proper subset.

$A \subseteq B$ is read as A is a subset of B.

Since every set is a subset of itself, the universal set is also a subset of itself.  

U $\subseteq$ U.

Example: Let the set of real numbers be the universal set. 

$U = \left\{ x | x\; \text{is a real number}\right\}$

We can list a few subsets as

U is a subset of itself.

Set of rational numbers is the subset of the set of real numbers.

Set of integers is the subset of the set of real numbers.

Set of natural numbers is the subset of the set of real numbers.

Set of whole numbers is the subset of the set of real numbers.

Facts about Universal Set

• Universal set is the largest set under consideration for a given context that includes all the possible elements for a given subject.
• The universal set is the superset of all the sets that can be defined for the subject under consideration.
• The complement of a universal set is always an empty set.
• The concept of Universal set is extremely useful to study the properties of sets, and also the relationship between different sets under a given context.

In this article, we have learned about the universal set, its complement and subsets. Let’s solve some universal set examples and practice problems for better understanding. 

1. Assuming P is the set of positive odd integers and Q is the set of positive even integers, write the universal set of P and Q.

Solution: 

The sets P and Q can be written as:

$P = \left\{1, 3, 9…\right\}$ 

$Q = \left\{2, 4, 6…\right\}$

The universal set will be the set of positive integers (or natural numbers ).

$U = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10…\right\}$

2. Draw a Venn diagram to represent the following sets:

$U = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9\right\}, A = \left\{1, 2, 5, 6\right\}, B = \left\{3, 9\right\}$

Solution:  

Draw a rectangle and label it U to represent the universal set. Draw two circles within the rectangle to represent the sets A and B. Label the circles and write the given elements in each circle . Write the remaining elements of the universal set outside the circles but within the rectangle.

$A = \left\{1, 2, 5, 6\right\}$ 

$B = \left\{3, 9\right\}$

The elements 4, 7, and 8 are not in the sets A and B. We will show them in the universal set.

3. State whether the following statements are true or false.

1) Every set is a subset of itself.

2) An empty set is not a subset of the universal set.

3) The universal is the complement of an empty set.

False 

An empty set is a subset of every set.

True. 

The universal set and empty set are complements of each other.

4. Observe the following Venn diagram and write down the universal set.

Set $A = \left\{3, 4, 5\right\}$

Set $B = \left\{1, 5\right\}$

5 is common between both A and B.

The elements 2 and 6 are outside the sets A and B. 

Thus, the universal set $U = \left\{1, 2, 3, 4, 5, 6\right\}$

Universal Set in Math - Definition, Symbol, Examples, Facts, FAQs

Attend this quiz & Test your knowledge.

When studying the population of the country USA, what would be the universal set?

An empty set is ________., let $a = \left\{1, 2\right\}, b = \left\{2, 4\right\}, c = \left\{4, 5, 6\right\}$. find the universal set for sets a, b, c., which of the following is a universal set for the equilateral triangle, if u is the universal set containing the set of all the continents, which of the following will be a subset of u.

What is a superset?

If set A is the subset of set B, then set B is called the superset of A. This means that set B has all the elements of set A. If B is superset of A, we write it as $B \supset A$.

What is the difference between the symbols $\subset$ and $\subseteq$ in set theory?

A $\subseteq$ B means that A is a subset of B.

A $\subset$ B means that A is a proper subset of B. Every element of A is an element of B, but A is not equal to B.

What is the formula to calculate the universal set?

There is no specific formula to find a universal set. The trick here is to represent all the elements in a set to get the universal set.

Can a universal set be an empty set ?

The complement of the universal set can be considered an empty set because as the universal set includes the set of all elements, the empty set will include no elements.

Is zero set a null set?

A zero set refers to a set with one element “0.” It is represented as $\left\{0\right\}$. Since this set has an element, it cannot be called a null set. It is a singleton set.

What is the difference between a universal set and an empty set?

A universal set includes all possible elements in each context, while an empty set contains no elements.

Can a universal set be finite?

Yes, a universal set can be finite, but it must include all possible elements in the given context.

How is a universal set related to a subset?

As the universal set includes all possible elements, every related set is a subset of the universal set.

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Universal Set

A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Say if A and B are two sets, such as A = {1,2,3} and B = {1,a,b,c}, then the universal set associated with these two sets is given by U = {1,2,3,a,b,c}.

In Mathematics, the collection of elements or group of objects is called a Set. There are various types of sets like Empty set, Finite set, Infinite set, Equivalent set, Subset, Superset and Universal set. All these sets have their own importance in Mathematics. There is a lot of usage of sets in our day-to-day life, but normally  they are used to represent bulk data or collection of data in a database. For example, our hand is a set of fingers, where each finger is different from the other one. The notation of set is usually given by curly brackets, {} and each element in the set is separated by commas like {4, 7, 9}, where 4, 7, and 9 are the elements of sets.

Universal set contains a group of objects or elements which are available in all the sets and is represented in a Venn diagram . 

Universal Set Definition

A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘U’.

Suppose Set A consists of all even numbers such that, A = {2, 4, 6, 8, 10, …} and set B consists of all odd numbers, such that, B = {1, 3, 5, 7, 9, …}. The universal set U consists of all natural numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}. Therefore, as we know, all the even and odd numbers are a part of natural numbers. Therefore, Set U has all the elements of Set A and Set B.

There is no formula to find the universal set, we just have to represent all the elements in a single which is collectively called a universal set.

Fact:  There is no standard notation for Universal set symbol, it can also be denoted by any other entity like ‘V’ or ‘ξ’.

Note:  I f Universal set contains Sets A, B and C, then these sets are also called subsets of Universal set. Denoted by;

A ⊂ U (A subset of U)

B ⊂ U (B subset of U)

C ⊂ U (C subset of U)

Venn Diagram of Universal Set

For Venn diagram representation of the universal set, we can take the example as;

U={heptagon} consisting of set A={pentagon, hexagon, octagon} and set C={nonagon}.

We can understand the concept of Universal set also by taking an example of the real world. In this world, we have set of a human being, set of animals and also set of all living things, which we can consider as a subset of U. But we cannot consider a set of trees as a subset of U.

Complement of Universal Set

There is a complement of set for every set. The empty set is defined as the complement of the universal set. That means where Universal set consists of a set of all elements, the empty set contains no elements of the subsets. The empty set is also called a Null set and is denoted by ‘{}’.

Universal set and Union of set

Students sometimes get confused between universal set and union of the set. They think both are the same but that is not true.

The universal set is a set which consists of all the elements or objects, including its own elements. It is represented by just a symbol ‘U’. But the union of sets is an operation performed on two sets, say A and B, which results in a set that has all elements belonging either to set A and set B or both. The operand of union of set is given by ‘∪’.

For example, Set A = {a,b,c} and set B={c,d,e} and U={1,2}. Therefore, the universal set for set A, B and U itself will be;

U ={a,b,c,d,e,1,2}

You can see, U is also a set which collectively shows all the elements of A, B and of itself.

Now, the union of set A and B is given by:

A∪B = {a,b,c,d,e}

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Set Theory: Universal Set

Related Pages More Lessons On Sets Subsets Venn Diagrams

In these lessons, we will learn what is a universal set and how it may be represented in a Venn Diagram.

What Is A Universal Set?

The following diagram explains what is a Universal Set and gives an example of a Universal Set. Scroll down the page if you need more explanations and examples about Universal Sets.

A universal set is the set of all elements under consideration, denoted by capital U or sometimes capital E.

Example: Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets. a) A = { x : x is a factor of 60} b) B = { x : x is a prime number}

Solution: The elements of sets A and B can only be selected from the given universal set U. a) A = {5, 6, 10, 12} b) B = {5, 7, 11}

In Venn diagrams, the universal set is usually represented by a rectangle and labeled U.

Example: Draw a Venn diagram to represent the following sets: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 6}, B = {3, 9}

Solution: Step 1: Draw a rectangle and label it U to represent the universal set.

Step 2: Draw circles within the rectangle to represent the other sets. Label the circles and write the relevant elements in each circle.

Step 3: Write the remaining elements outside the circles but within the rectangle.

The Universal Set And Set Complements

Example: Let the universal set, U = {a, e, i, o, u} Let the subset A = {a, e} Then the complement of set A, A’ = {i, o, u}

Intersection, Union And Complement Of Sets

A ∩ B (read as A intersection B) are members that are common to both set A and set B. A ∪ B (read as A union B) are members that are in set A or set B or both. A’ (read as A complement) are members that are not in set A.

What Is Universal Set And Absolute Complement?

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Universal Set

The universal set is a set that contains all the elements related to a specific context. The universal set is denoted by U which is a superset of all sets with respect to a given context. For example, in the context set A represents the set of vowels, set B represents the set of consonants, set C represents the set of alphabets made up of straight lines only, etc, the universal set can be the set of all alphabets. 

The universal set has important applications in probability where it is used to represent the sample space. Let us learn more about universal set along with its Venn diagram, its properties, and examples.

What Is Universal Set?

A universal set is a collection of all elements or members of all the related sets, known as its subsets. The set of all real numbers is the universal set in the context of sets of rational numbers , irrational numbers , integers , whole numbers , natural numbers , etc. In a particular context:

• Universal set is the superset of all sets.
• All sets are subsets of universal set.

Universal Set Definition

The universal set is the set of all elements or members of all related sets. It is usually denoted by the symbol E or U. For example, in human population studies, the universal set is the set of all the people in the world. The set of all people in each country can be considered as a subset of this universal set.

• A universal set can be either a finite or infinite set.
• The set of natural numbers is a typical example of an infinite universal set.

Symbol of Universal Set

The universal set is represented by the symbol E or U. It consists of all the elements of its subsets, along with some extra elements.

Example of Universal Set

Let's consider an example with three sets, A, B, and C. Here, A = {2, 4, 6}, B = {1, 3, 7, 9, 11}, and C = {4, 8, 11}. We need to find the universal set for all three sets A, B, and C. All the elements of the given sets are contained in the universal set. Thus, the universal set U of A, B, and C can be given by U = {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12}.

We can see that all the elements of the three sets are present in the universal set without any repetition. Thus, we can say that all the elements in the universal set are unique. The sets A, B, and C are contained in the universal set, then these sets are also called subsets of the Universal set.

• A ⊂ U (A is the subset of U)
• B ⊂ U (B is the subset of U)
• C ⊂ U (C is the subset of U)

Complement of Universal Set

For a subset A of the universal set (U), its complement is represented as A' which includes the elements of the universal set but not the elements of set A. The universal set consists of a set of all elements of all its related subsets, whereas the empty set contains no elements of the subsets. Thus, the complement of the universal set is an empty set ( null set ), denoted by ‘{}’ or the symbol 'Φ'.

Venn Diagram of Universal Set

Most of the time we use the Venn diagram to show the relationship between sets. Venn diagrams are the graphical representation of the sets. The universal set is represented by a rectangle and its subsets are represented by circles. Look at the Venn diagram shown below.

Here, we can see that the universal set U has these elements {1, 2, 3, ..., 10}, and the subset of this universal set is the set A = {2, 4, 6, 8, 10}.

Difference Between Universal Set and Union of Sets

Usually, students have confusion in differentiating between the union of sets and the universal set. We can understand the difference better by looking at their definitions.

Thus, the universal set is the union of all its subsets along with some extra elements in some cases. The following example can be used to understand this difference better. Consider three sets with elements set A = {a, b, c} and set B = {e, f, g}. Let's find the universal set U and the union of sets A and B.

• The universal set of the above two sets can be the set of all alphabets {a, b, c, d, …, z}
• The union between A and B is given as: A ∪ B = {a, b, c, e, f, g}

Thus, we can see that the universal set contains the elements from A, B, and U itself, whereas the union of A and B contains elements from only A and B.

Important Notes on Universal Set:

• If A is a subset of the universal set U, then all the elements in U that are not in A are called the complement of A (denoted by A').
• If A is a subset of the universal set U, then the complement of A is also a subset of U.
• The complement of a universal set is always an empty set.
• A set and its complement are the disjoint sets .

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• Finite and Infinite Sets
• Sets Formula
• Intersection of Sets
• Operation on Sets

Universal Set Examples

Example 1: Given below is a Venn diagram representing the sets, A and B. Determine the elements of the universal set for its given subsets, A and B.

We know that any universal set is represented by a rectangle and its subsets are represented by circles.

Here, the subset A = {3, 7, 9} and the subset B = {4, 8}. Clearly, A and B are disjoint sets because they have no common element. Also, the elements that are not contained in A and B are contained in the universal set.

Thus, the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

Answer:  ∴ The universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

Example 2: Consider the universal set U = {2, 4, 5, 14, 17, 28, 35, 52}. List the elements of the following sets:

i) A = {x: x is a factor of 10}

ii) B = {x: x is a multiple of 14}

Given: The universal set U = {2, 4, 5, 14, 17, 28, 35, 52}

Consider the given universal set U = {2, 4, 5, 14, 17, 28, 35, 52}.

The factors of 10 contained in this set are 2 and 5 and the multiples of 14 in this set are 14 and 28.

Thus, the set A = {2, 5} and set B = {14, 28}.

Answer:  ∴ For U = {2, 4, 5, 14, 17, 28, 35, 52}, set A = {2, 5} and set B = {14, 28}.

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Example 3:  Which of the following set(s) can be the universal set with the related sets as A = {1, 2, 3, 5}, B = {11, 13}, and C = {20, 21, 22}? Select all that apply.

(a) U = {1, 2, 3, …, 10} (b) U = {1, 2, 3, …, 50} (c) U = {1, 2, 3, …, 25} (d) U = {1, 2, 3, …, 20}

The given sets are A = {1, 2, 3, 5}, B = {11, 13}, and C = {20, 21, 22}.

The universal set in this context should contain all the elements of A, B, and C. Also, it may contain some different elements as well.

Using this definition, among the given options, only (b) U = {1, 2, 3, …, 50} (or) (c) U = {1, 2, 3, …, 25} can be the universal sets.

Answer:  Options (b) and (c).

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Practice Questions on Universal Set

Faqs on universal set, what is the universal set in math.

The universal set is the set of all elements or members of all the related sets. The universal set is usually denoted by the symbol E or U. For example, for the set of all kinds of prisms, the universal set is the set of all three-dimensional shapes.

What Are Universal Sets and Subsets?

If all the elements of set A are also the elements of another set B, then we can say that A is the subset of B. Then, the subsets can actually be created from any given universal set. We should also note that any universal set is actually a subset of itself. But the elements in a subset are less than the elements in the universal set from which the subset is created.

How To Represent Universal Set in a Venn Diagram?

Most of the time we use the Venn diagram to show the relationship between sets for more clarity. Venn diagrams are the graphical representation of the sets in which the universal set is represented by rectangles and its subsets are represented by circles.

What Is the Complement of the Universal Set?

The complement of the universal set can be considered as an empty set because when the universal set contains the set of all elements, then the empty set will contain no elements of the subsets. The null set is another term used for the empty set, and it is denoted by the symbol '{}'.

How Do You Find Universal Sets?

We cannot define a unique universal set of some of its subsets are given. Because the universal set may contain some elements which none of its subsets have. Usually, a universal set is already defined in a context and we are not asked to determine it.

What Is the Difference Between a Universal Set and a Union of Seta?

The universal set is defined as a set containing all elements or members of all the related sets, known as its subsets, whereas the union of sets is one of the set operations between two sets where the resultant set contains all the elements of the given sets.

What Is the Universal Set of All Right Triangles?

All triangles have three sides and three angles. There are different types of triangles based on their sides and angles. Thus, the universal set of all right triangles can be

• the set containing all the polygons or
• the set of all triangles , etc.

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• Universal set – Definition and Examples

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What is a Universal Set?

How to represent a universal set, difference between universal set and union of set, representation of universal set through venn diagram , universal set and its subsets, practice problems , universal set – definition and examples.

In this article, we will be developing an understanding of the Universal Set.

A universal set (usually denoted by U) is a set that has elements of all the related sets without any repetition of elements.  

We will cover the following topics in this article:

• What is a universal set?
• How to represent the universal set?
• Difference Between Universal Set and Union of Set.
• Practice Problems  

The universal set is the reflection of its name. A universal set, in general terms, is defined as the set of all objects under consideration. A universal set is a set that contains all the elements or objects of other sets, including its elements. It is typically denoted by “U.” 

The exact definition of U is based on the context or theory being considered. The elements contained in U may vary according to the context of sets being stated. For instance, we might describe U as the collection of all living things on planet earth. In this case, the set of all the mammals, set of all the reptiles, and set of all the birds become a subset of U. Similarly, we can say that set of all countries in the world is universal.

In mathematical terms, the universal set is the set of whole numbers. Depending on the context, this set may also be the set of all the shapes, a set of all the second-degree polynomials, and many more. 

Since a set is a collection of various entities with a common property, the universal set is the collection of all the elements relevant to a specific subject. It could be anything like real numbers, a deck of cards, the world’s inhabitants, etc. Elements in a universal set are not repeated elements, so all these elements are distinct and unique. 

Let’s solve an example to develop a firm understanding of universal sets.

Consider, there are three sets, namely X, Y, and Z. The elements of each set are given below:

X={2, 4, 6, 8}

Y={3, 7, 9, 11}

Z={4, 8, 11}

Find the universal set for all three sets X, Y, and Z.

We know that the universal set contains all the elements of the given sets; thus, the universal set of X, Y, and Z is given by:

U={2, 3, 4, 6, 7, 8, 9, 11}

We can see from the solution above that the elements of sets X, Y, and Z are completely available in the Universal set ‘U’. Also, there are no elements repeated in the universal set, and all the elements are unique.

Now that we know what a universal set is, the next topic addresses its representation. Generally, the universal set is represented by the capital letter “U.” Their representation is the same as any other set. All the elements of the universal set are enclosed in the ‘{}’ curly brackets. 

A universal set can be both finite or infinite. If the universal set is infinite, or even if it’s finite with a huge number of elements, then the ellipses, the three dots (…), are used to represent the universal set. An example is shown below:

U = {1, 2, 3, …}

U = {1, 2, 3, …, 4000}

In mathematical terms, the universal set is a collection of numbers. So we can also use bold capital letters for its representation. There is no specific universal set symbol notation.

Consider a universal set of natural numbers, which is represented by N. The notation of this universal set is somewhat like follows:  

N = {1, 2, 3, 4, …}

Some examples of the universal set are given below.

Let A is equal to a set of odd numbers, and B is equal to the set of even numbers. What is the universal set of A and B?

The sets A and B can be written as:

A = {1, 3, 5, 7, 9…}

B = {2, 4, 6, 8, 10…}

So, the universal set is:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10…}

From the above example, we can see that the universal set U of the sets A and B results in the set of natural numbers.

Students mostly get confused between the universal set and the union of the set. Some make the common mistake of thinking that the sets are similar. But that is not the case. We can understand this better by analyzing the definitions of these two sets. 

The universal set, including its elements, is a set composed of all elements or objects. Whereas the union of sets is a two-set operation, say A and B, resulting in a set with all the elements that belong to both sets A and B.   

The following example will help us understand this difference better. 

Consider three sets with elements U = {3, 5}, set X = {a, s, d}, and set Y = {e, f, g}. Find the universal set U and the union of sets X and Y?

The universal set of the 3 sets is given as follows:

U = { a, s, d, e, f, g, 3, 5}

Now, let’s find the union. The union between X and Y is given as:

X U Y = {a, s, d, e, h, g}

Thus, the universal set is the combination of X, Y, and U itself, whereas the union of X and Y contains X and Y elements only.

The Venn Diagram represents two or more circles used to illustrate the relationships among different sets of finite objects.  Since the set representation is entirely visual, so different shapes are used to depict Venn diagrams.

In the Venn diagram, the circles are used to represent sets. These circles or sets are contained within the universal set, which is represented by a rectangle. 

The universal set is represented by a rectangle in the Venn diagram and denoted by U. Let’s consider an example. Suppose a set of multiples of 2 is given as A = {2, 4, 20, 22}. Now consider another set B which is the set of even numbers given as B = {2, 4, 6, 8, 10}. The universal set in this case is the set of numbers U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The Venn diagrams as follows:

Let’s solve an example to understand this better. 

Let set A = {1, 3, 5, 7, 9, 11} and set B = {x : x is a primary number and 2<x<17}. Show the relationship between sets A and B where universal set U is U = {set of natural numbers till 20}.

To solve this example, let’s first simplify the sets.

Set B can be rewritten as:

B = {3, 5, 7, 11, 13}

Similarly, set U can be rewritten as:

U = {1, 2, 3, 4, 5, …, 20} 

Their relationship through the Venn diagram is as follows:

Since a universal set is a set with all the elements related to any specified context of sets, all the other sets in it are the Universal set subsets. 

Before we deal with the universal set and its subsets, let’s have an overview of subsets. A set A is a subset of B if all A’s elements are also B’s elements.  The symbol represents the subset ⊆.

Thus we represent the A as a subset of B as:

Since all the sets are the subset of themselves, hence, the universal set is also the subset of itself which can be represented as:

Consider the example below to evaluate this concept.

Consider three sets U = {0,1, 2, 3, 4, 5…}, X = {1, 3, 5}, and Y = {2, 4, 6}. List the possible subsets.

It is evident that set X and Y are the subsets of U, so:

X is the proper subset of universal set U.

Similarly, for the set Y:

Y is the proper subset of universal set U.

Some other possible subsets obtained from the universal set can be:

O = {1, 3, 5, 7,…} i.e. a set of all odd numbers.

E = {2, 4, 6, 8,…} i.e. a set of all even numbers.

To further strengthen the concept of Venn diagrams, consider the following practice problems.

• If X = {Asia, Africa, North America}, and Y = {South America, Antarctica, Europe, Australia}, what could be the universal set?
• Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets. 

         (a) A = {x : x is a factor of 60}       (b) B = {x : x is a prime number}

      3. If U = {2, 4, 6, 8, 10, 12, 14}, then which of the following are subsets of U. 

          X = {2, 4} , Y = {0} , Z = {1, 10, 5, 12} , A = {5, 14, 1}  , B = {2, 4, 6, 8}

       4. If A = { x : x = 2n+1,  n ∈ N}

                B = { x : x = 2n,  n ∈ N}

               C = { x : x = n,  n ∈ N}

          Identify the universal set among A, B and C.

      5. Consider four sets: 

          U = {1, 3, 5, 7, 9, 13…}

         Which one of the following is the subset of U?

         X = {1, 3, 5}

         Y = {2, 4, 6}

        Z = { 1, 8, 13}

• U = {Continents}
• a) A = {5, 6, 10, 12};        b) B = {5, 7, 11}

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Introduction to Sets

Forget everything you know about numbers.

In fact, forget you even know what a number is.

This is where mathematics starts.

Instead of math with numbers, we will now think about math with "things".

What is a set? Well, simply put, it's a collection .

First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.

For example, the items you wear: hat, shirt, jacket, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set .

So it is just things grouped together with a certain property in common.

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".

This is the notation for the two previous examples:

{socks, shoes, watches, shirts, ...} {index, middle, ring, pinky}

Notice how the first example has the "..." (three dots together).

The three dots ... are called an ellipsis, and mean "continue on".

So that means the first example continues on ... for infinity.

(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)

• The first set {socks, shoes, watches, shirts, ...} we call an infinite set ,
• the second set {index, middle, ring, pinky} we call a finite set .

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters:

{a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?)

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

And so on. We can come up with all different types of sets.

We can also define a set by its properties, such as {x|x>0} which means "the set of all x's, such that x is greater than 0", see Set-Builder Notation to learn more.

And we can have sets of numbers that have no common property, they are just defined that way. For example:

Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets .

Universal Set

Some more notation.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Example: Are A and B equal where:

• A is the set whose members are the first four positive whole numbers
• B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!

And the equals sign (=) is used to show equality, so we write:

Example: Are these sets equal?

• A is {1, 2, 3}
• B is {3, 1, 2}

Yes, they are equal!

They both contain exactly the members 1, 2 and 3.

It doesn't matter where each member appears, so long as it is there.

When we define a set, if we take pieces of that set, we can form what is called a subset .

Example: the set {1, 2, 3, 4, 5}

A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.

In general:

A is a subset of B if and only if every element of A is in B.

So let's use this definition in some examples.

Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, and 1 is in B as well. So far so good.

3 is in A and 3 is also in B.

4 is in A, and 4 is in B.

That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2 . Is A a subset of B? And is B a subset of A?

Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.

The sets are:

• A = {..., −8, −4, 0, 4, 8, ...}
• B = {..., −8, −6, −4, −2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:

A is a subset of B, but B is not a subset of A

Proper Subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.

Let A be a set. Is every element of A in A ?

Well, umm, yes of course , right?

So that means that A is a subset of A . It is a subset of itself!

This doesn't seem very proper , does it? If we want our subsets to be proper we introduce (what else but) proper subsets :

A is a proper subset of B if and only if every element of A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

Notice that when A is a proper subset of B then it is also a subset of B.

Even More Notation

When we say that A is a subset of B, we write A ⊆ B

Or we can say that A is not a subset of B by A ⊈ B

When we talk about proper subsets, we take out the line underneath and so it becomes A ⊂ B or if we want to say the opposite A ⊄ B

Empty (or Null) Set

This is probably the weirdest thing about sets.

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set with no elements .

This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.

It is represented by ∅

Or by {} (a set with no elements)

Some other examples of the empty set are the set of countries south of the south pole .

So what's so weird about the empty set? Well, that part comes next.

Empty Set and Subsets

So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?

Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A . But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.

A good way to think about it is: we can't find any elements in the empty set that aren't in A , so it must be that all elements in the empty set are in A.

So the answer to the posed question is a resounding yes .

The empty set is a subset of every set, including the empty set itself.

No, not the order of the elements. In sets it does not matter what order the elements are in .

Example: {1,2,3,4} is the same set as {3,1,4,2}

When we say order in sets we mean the size of the set .

Another (better) name for this is cardinality .

A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).

For finite sets the order (or cardinality) is the number of elements .

Example: {10, 20, 30, 40} has an order of 4.

For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.

Arg! Not more notation!

Nah, just kidding. No more notation.

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4.3: Unions and Intersections

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• Page ID 24953

• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

We can form a new set from existing sets by carrying out a set operation.

Definition: \(A \cap B\)

Given two sets \(A\) and \(B\), define their intersection to be the set

\[A \cap B = \{ x\in{\cal U} \mid x \in A \wedge x \in B \}\]

Loosely speaking, \(A \cap B\) contains elements common to both \(A\) and \(B\).

Definition: \(A \cup B\)

The union of \(A\) and \(B\) is defined as

\[A \cup B = \{ x\in{\cal U} \mid x \in A \vee x \in B \}\]

Thus \(A \cup B\) is, as the name suggests, the set combining all the elements from \(A\) and \(B\).

        UNION

Definition: \(A-B\)

The set difference \(A-B\), sometimes written as \(A \setminus B\), is defined as

\[A- B = \{ x\in{\cal U} \mid x \in A \wedge x \not\in B \}\]

In words, \(A-B\) contains elements that can only be found in \(A\) but not in \(B\). Operationally speaking, \(A-B\) is the set obtained from \(A\) by removing the elements that also belong to \(B\). 

Definition: \(\overline{A}\)

The complement of \(A\), denoted by \(\overline{A}\), \(A'\) or \(A^c\), is defined as 

\[\overline{A} = \{ x\in{\cal U} \mid x \notin A \}\]

Definition: \(A \bigtriangleup B\)

The symmetric difference   \(A \bigtriangleup B\), is defined as

\[A \bigtriangleup B = (A - B) \cup (B - A)\] 

Definition: Disjoint

Two sets are disjoint if their intersection is empty.

For example, consider \(S=\{1,3,5\}\) and \(T=\{2,8,10,14\}\).

\(S \cap T = \emptyset\) so \(S\) and \(T\) are disjoint.

We would like to remind the readers that it is not uncommon among authors to adopt different notations for the same mathematical concept. Likewise, the same notation could mean something different in another textbook or even another branch of mathematics. It is important to develop the habit of examining the context and making sure that you understand the meaning of the notations when you start reading a mathematical exposition.

Example \(\PageIndex{1}\label{eg:unionint-01}\)

Let \({\cal U}=\{1,2,3,4,5\}\), \(A=\{1,2,3\}\), and \(B=\{3,4\}\). Find \(A\cap B\), \(A\cup B\), \(A-B\), \(B-A\), \(A\bigtriangleup B\), \(\overline{A}\), and \(\overline{B}\).

We have \[\begin{aligned} A\cap B &=& \{3\}, \\ A\cup B &=& \{1,2,3,4\}, \\ A - B &=& \{1,2\}, \\ B \bigtriangleup A &=& \{1,2,4\}. \end{aligned}\] We also find \(\overline{A} = \{4,5\}\), and \(\overline{B} = \{1,2,5\}\).

hands-on exercise \(\PageIndex{1}\label{he:unionint-01}\)

Let \({\cal U} = \{\mbox{John}, \mbox{Mary}, \mbox{Dave}, \mbox{Lucy}, \mbox{Peter}, \mbox{Larry}\}\), \[A = \{\mbox{John}, \mbox{Mary}, \mbox{Dave}\}, \qquad\mbox{and}\qquad B = \{\mbox{John}, \mbox{Larry}, \mbox{Lucy}\}.\] Find \(A\cap B\), \(A\cup B\), \(A-B\), \(B-A\), \(\overline{A}\), and \(\overline{B}\).

hands-on exercise \(\PageIndex{2}\label{he:unionint-02}\)

If \(A\subseteq B\), what would be \(A-B\)?

Example \(\PageIndex{2}\label{eg:unionint-02}\)

The set of integers can be written as the \[\mathbb{Z} = \{-1,-2,-3,\ldots\} \cup \{0\} \cup \{1,2,3,\ldots\}.\] Can we replace \(\{0\}\) with 0? Explain.

hands-on exercise \(\PageIndex{3}\label{he:unionint-03}\)

Explain why the following expressions are syntactically incorrect.

• \(\mathbb{Z} = \{-1,-2,-3,\ldots\} \cup \;0\; \cup \{1,2,3,\ldots\}\).
• \(\mathbb{Z} = \ldots,-3,-2,-1 \;\cup\; 0 \;\cup\; 1,2,3,\ldots\,\)
• \(\mathbb{Z} = \ldots,-3,-2,-1 \;+\; 0 \;+\; 1,2,3,\ldots\,\)
• \(\mathbb{Z} = \mathbb{Z} ^- \;\cup\; 0 \;\cup\; \mathbb{Z} ^+\)

How would you fix the errors in these expressions?

Example \(\PageIndex{3}\label{eg:unionint-03}\)

For any set \(A\), what are \(A\cap\emptyset\), \(A\cup\emptyset\), \(A-\emptyset\), \(\emptyset-A\) and \(\overline{\overline{A}}\)?

It is clear that \[A\cap\emptyset = \emptyset, \qquad A\cup\emptyset = A, \qquad\mbox{and}\qquad A-\emptyset = A.\] From the definition of set difference, we find \(\emptyset-A = \emptyset\). Finally, \(\overline{\overline{A}} = A\).

Example \(\PageIndex{4}\label{eg:unionint-04}\)

Write, in interval notation, \([5,8)\cup(6,9]\) and \([5,8)\cap(6,9]\).

The answers are \[[5,8)\cup(6,9] = [5,9], \qquad\mbox{and}\qquad [5,8)\cap(6,9] = (6,8).\] They are obtained by comparing the location of the two intervals on the real number line.

hands-on exercise \(\PageIndex{4}\label{he:unionint-04}\)

Write, in interval notation, \((0,3)\cup[-1,2)\) and \((0,3)\cap[-1,2)\).

Example \(\PageIndex{5}\label{eg:unionint-05}\)

We are now able to describe the following set \[\{x\in\mathbb{R} \mid (x<5) \vee (x>7)\}\] in the interval notation. It can be written as either \((-\infty,5)\cup(7,\infty)\) or, using complement, \(\mathbb{R}-[5,7\,]\). Consequently, saying \(x\notin[5,7\,]\) is the same as saying \(x\in(-\infty,5) \cup(7,\infty)\), or equivalently, \(x\in \mathbb{R}-[5,7\,]\).

To Prove a Set is Empty

To  prove a set is empty , use a proof by contradiction with these steps:

(1) Assume not.  That, is assume \(\ldots\) is not empty.

(2) This means there is an element is \(\ldots\) by definition of the empty set. 

(3) Let \(x \in \ldots \).

(4) Come to a contradition and wrap up the proof.

Example \(\PageIndex{6}\)

Prove: \(\forall A \in {\cal U}, A \cap \emptyset = \emptyset.\)

Proof:  Assume not. That is, assume for some set \(A,\)  \(A \cap \emptyset \neq \emptyset.\) By definition of the empty set, this means there is an element in \(A \cap \emptyset .\)

Let \(x \in A \cap \emptyset .\)

\(x \in A \wedge x\in \emptyset\) by definition of intersection.

This says \(x \in \emptyset \), but the empty set has no elements!  This is a contradiction!

Thus, our assumption is false, and the original statement is true. \(\forall A \in {\cal U}, A \cap \emptyset = \emptyset.\)

Set Properties

(a) These properties should make sense to you and you should be able to prove them.  However, you are not to use them as reasons in a proof.  Rather your justifications for steps in a proof need to come directly from definitions. The exception to this is DeMorgan's Laws which you may reference as a reason in a proof.

(b) You do not need to memorize these properties or their names.  However, you should know the meanings of: commutative, associative and distributive.  Also, you should know DeMorgan's Laws by name and substance.

The following properties hold for any sets \(A\), \(B\), and \(C\) in a universal set \({\cal U}\).

• Commutative properties : \(\begin{array}[t]{l} A \cup B = B \cup A, \\ A \cap B = B \cap A. \end{array}\)
• Associative properties : \(\begin{array}[t]{l} (A \cup B) \cup C = A \cup (B \cup C), \\ (A \cap B) \cap C = A \cap (B \cap C). \end{array}\)
• Distributive laws : \(\begin{array}[t]{l} A \cup (B \cap C) = (A \cup B) \cap (A \cup C), \\ A \cap (B \cup C) = (A \cap B) \cup (A \cap C). \end{array}\)
• Idempotent laws : \(\begin{array}[t]{l} A \cup A = A, \\ A \cap A = A. \end{array}\)
• De Morgan’s laws : \(\begin{array}[t]{l} \mbox{  (a)  } \overline{A \cup B} = \overline{A} \cap \overline{B}, \\ \mbox{  (b)  }  \overline{A \cap B} = \overline{A} \cup \overline{B}. \end{array}\)
• Laws of the excluded middle , or inverse laws : \(\begin{array}[t]{l} A \cup \overline{A} = {\cal U}, \\ A \cap \overline{A} = \emptyset. \end{array}\)

As an illustration, we shall prove the distributive law \[A \cup (B \cap C) = (A \cup B) \cap (A \cup C).\]  

We need to show that \[A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C), \qquad\mbox{and}\qquad (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C).\]

Here is a proof of the distributive law \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).

hands-on exercise \(\PageIndex{5}\label{he:unionint-05}\)

Prove that \(A\cap(B\cup C) = (A\cap B)\cup(A\cap C)\).

hands-on exercise \(\PageIndex{6}\label{he:unionint-06}\)

Prove that if \(A\subseteq B\) and \(A\subseteq C\), then \(A\subseteq B\cap C\).

Let us start with a draft. The statement we want to prove takes the form of \[(A\subseteq B) \wedge (A\subseteq C) \Rightarrow A\subseteq B\cap C.\] Hence, what do we assume and what do we want to prove?

Did you put down we assume \(A\subseteq B\) and \(A\subseteq C\), and we want to prove \(A\subseteq B\cap C\)? Great! Now, what does it mean by \(A\subseteq B\)? How about \(A\subseteq C\)? What is the meaning of \(A\subseteq B\cap C\)?

How can you use the first two pieces of information to obtain what we need to establish?

Now it is time to put everything together, and polish it into a final version. Remember three things:

• the outline of the proof,
• the reason in each step of the main argument, and
• the introduction and the conclusion.

Put the complete proof in the space below.

Here are two results involving complements.

Theorem \(\PageIndex{1}\label{thm:subsetsbar}\)

For any two sets \(A\) and \(B\), we have \(A \subseteq B \Leftrightarrow \overline{B} \subseteq \overline{A}\).

Theorem \(\PageIndex{2}\label{thm:genDeMor}\)

For any sets \(A\), \(B\) and \(C\),  

(a) \(A-(B \cup C)=(A-B) \cap (A-C)\)

(b) \(A-(B \cap C)=(A-B) \cup (A-C)\)

Summary and Review

• Memorize the definitions of intersection, union, and set difference. We rely on them to prove or derive new results.
• The intersection of two sets \(A\) and \(B\), denoted \(A\cap B\), is the set of elements common to both \(A\) and \(B\). In symbols, \(\forall x\in{\cal U}\,\big[x\in A\cap B \Leftrightarrow (x\in A \wedge x\in B)\big]\).
• The union of two sets \(A\) and \(B\), denoted \(A\cup B\), is the set that combines all the elements in \(A\) and \(B\). In symbols, \(\forall x\in{\cal U}\,\big[x\in A\cup B \Leftrightarrow (x\in A\vee x\in B)\big]\).
• The set difference between two sets \(A\) and \(B\), denoted by \(A-B\), is the set of elements that can only be found in \(A\) but not in \(B\). In symbols, it means \(\forall x\in{\cal U}\, \big[x\in A-B \Leftrightarrow (x\in A \wedge x\notin B)\big]\).
• The symmetric difference between two sets \(A\) and \(B\), denoted by \(A \bigtriangleup B\), is the set of elements that can be found in \(A\) and in \(B\), but not in both \(A\) and \(B\).  In symbols, it means \(\forall x\in{\cal U}\, \big[x\in A \bigtriangleup B \Leftrightarrow x\in A-B  \vee x\in B-A)\big]\).

Exercises 

Exercise \(\PageIndex{1}\label{ex:unionint-01}\)

Write each of the following sets by listing its elements explicitly.

(a) \([-4,4]\cap\mathbb{Z}\)

(b) \((-4,4]\cap\mathbb{Z}\)

(c) \((-4,\infty)\cap\mathbb{Z}\)

(d) \((-\infty,4]\cap\mathbb{N}\)

(e) \((-4,\infty)\cap\mathbb{Z}^-\)

(f) \((4,5)\cap\mathbb{Z}\)  

(a) \(\{-4,-3,-2,-1,0,1,2,3,4\}\)

(b) \(\{-3,-2,-1,0,1,2,3,4\}\)

(c) \(\{-3,-2,-1,0,1,2,3,\ldots\}\)

Exercise \(\PageIndex{2}\label{ex:unionint-02}\)

Assume \({\cal U} = \mathbb{Z}\), and let

\(A=\{\ldots, -6,-4,-2,0,2,4,6, \ldots \} = 2\mathbb{Z},\)

\(B=\{\ldots, -9,-6,-3,0,3,6,9, \ldots \} = 3\mathbb{Z},\)

\(C=\{\ldots, -12,-8,-4,0,4,8,12, \ldots \} = 4\mathbb{Z}.\)

 Describe the following sets by listing their elements explicitly.

(a) \(A\cap B\)

(b) \(C-A\)

(c) \(A-B\)

(d) \(A\cap\overline{B}\)

(e) \(B-A\)

(f) \(B\cup C\)

(g) \((A\cup B)\cap C\)

(h) \((A\cup B)-C\)

Exercise \(\PageIndex{3}\label{ex:unionint-03}\)

Are these statements true or false?

(a) \([1,2]\cap[2,3] = \emptyset\)

(b) \([1,2)\cup(2,3] = [2,3]\)

(a) false (b) false

Exercise \(\PageIndex{4}\label{ex:unionint-04}\)

Let the universal set \({\cal U}\) be the set of people who voted in the 2012 U.S. presidential election. Define the subsets \(D\), \(B\), and \(W\) of \({\cal U}\) as follows: \[\begin{aligned} D &=& \{x\in{\cal U} \mid x \mbox{ registered as a Democrat}\}, \\ B &=& \{x\in{\cal U} \mid x \mbox{ voted for Barack Obama}\}, \\ W &=& \{x\in{\cal U} \mid x \mbox{ belonged to a union}\}. \end{aligned}\] Express the following subsets of \({\cal U}\) in terms of \(D\), \(B\), and \(W\).

(a) People who did not vote for Barack Obama.

(b) Union members who voted for Barack Obama.

(c) Registered Democrats who voted for Barack Obama but did not belong to a union.

(d) Union members who either were not registered as Democrats or voted for Barack Obama.

(e) People who voted for Barack Obama but were not registered as Democrats and were not union members.

(f) People who were either registered as Democrats and were union members, or did not vote for Barack Obama.

Exercise \(\PageIndex{5}\label{ex:unionint-05}\)

An insurance company classifies its set \({\cal U}\) of policy holders by the following sets: \[\begin{aligned} A &=& \{x\mid x\mbox{ drives a subcompact car}\}, \\ B &=& \{x\mid x\mbox{ drives a car older than 5 years}\}, \\ C &=& \{x\mid x\mbox{ is married}\}, \\ D &=& \{x\mid x\mbox{ is over 21 years old}\}, \\ E &=& \{x\mid x\mbox{ is a male}\}. \end{aligned}\] Describe each of the following subsets of \({\cal U}\) in terms of \(A\), \(B\), \(C\), \(D\), and \(E\).

(a) Male policy holders over 21 years old.

(b) Policy holders who are either female or drive cars more than 5 years old.

(c) Female policy holders over 21 years old who drive subcompact cars.

(d) Male policy holders who are either married or over 21 years old and do not drive subcompact cars.

(a) \(E\cap D\) (b) \(\overline{E}\cup B\)

Exercise \(\PageIndex{6}\label{ex:unionint-06}\)

Let \(A\) and \(B\) be arbitrary sets. Complete the following statements.

(a) \(A\subseteq B \Leftrightarrow A\cap B = \)        ___________________

(b) \(A\subseteq B \Leftrightarrow A\cup B = \)        ___________________

(c) \(A\subseteq B \Leftrightarrow A - B = \)           ___________________

(d) \(A\subset B \Leftrightarrow (A-B= \)  ___________________\(\wedge\,B-A\neq\) ___________________ \()\)

(e) \(A\subset B \Leftrightarrow (A\cap B=\)  ___________________\(\wedge\,A\cap B\neq\) ___________________  \()\)

(f) \(A - B = B - A \Leftrightarrow \)           ___________________

Exercise \(\PageIndex{7}\label{ex:unionint-07}\)

Give examples of sets \(A\) and \(B\) such that \(A\in B\) and \(A\subset B\).

For example, take \(A=\{x\}\), and \(B=\{\{x\},x\}\).

Exercise \(\PageIndex{8}\label{ex:unionint-08}\)

(a) Prove De Morgan’s law, (a) .

(b) Prove De Morgan’s law, (b) .

Exercise \(\PageIndex{9}\label{ex:unionint-09}\)

Let \(A\), \(B\), and \(C\) be any three sets. Prove that if \(A\subseteq C\) and \(B\subseteq C\), then \(A\cup B\subseteq C\).

Assume \(A\subseteq C\) and \(B\subseteq C\), we want to show that \(A\cup B \subseteq C\).

Let \(x\in A\cup B\). we want to show that \(x\in C\) as well.

Since \(x\in A\cup B\), then either \(x\in A\) or \(x\in B\) by definition of union.

Case 1: If \(x\in A\), then \(A\subseteq C\) implies that \(x\in C\) by definition of subset.

Case 2: If \(x\in B\), then \(B\subseteq C\) implies that \(x\in C\) by definition of subset.

In both cases, we find \(x\in C\). So, if \(x\in A\cup B\) then \(x\in C\).

This proves that \(A\cup B\subseteq C\) by definition of subset.

\(\therefore\) For any sets \(A\), \(B\), and \(C\) if \(A\subseteq C\) and \(B\subseteq C\), then \(A\cup B\subseteq C\).

Exercise \(\PageIndex{10}\label{ex:unionint-10}\)

Prove Theorem 4.3.1

Exercise \(\PageIndex{11}\label{ex:unionint-11}\)

(a) Prove Theorem 4.3.2 part (a)

(b) Prove Theorem 4.3.2 part (b)

Exercise \(\PageIndex{12}\label{ex:unionint-12}\)

Let \(A\), \(B\), and \(C\) be any three sets. Prove that

(a) \(A-B=A\cap\overline{B}\)

(b) \(A=(A-B)\cup(A\cap B)\)

(c) \(A-(B-C) = A\cap(\overline{B}\cup C)\)

(d) \((A-B)-C = A-(B\cup C)\)

Exercise \(\PageIndex{13}\label{ex:unionint-13}\)

Comment on the following statements. Are they syntactically correct?

(a) \(x\in A \cap x\in B \equiv x\in A\cap B\)

(b) \(x\in A\wedge B \Rightarrow x\in A\cap B\)

(a) The notation \(\cap\) is used to connect two sets, but “\(x\in A\)” and “\(x\in B\)” are both logical statements. We should also use \(\Leftrightarrow\) instead of \(\equiv\). The statement should have been written as “\(x\in A \,\wedge\, x\in B \Leftrightarrow x\in A\cap B\).”

(b) If we read it aloud, it sounds perfect: \[\mbox{If $x$ belongs to $A$ and $B$, then $x$ belongs to $A\cap B$}.\] The trouble is, every notation has its own meaning and specific usage. In this case, \(\wedge\) is not exactly a replacement for the English word “and.” Instead, it is the notation for joining two logical statements to form a conjunction. Before \(\wedge\), we have “\(x\in A\),” which is a logical statement. But, after \(\wedge\), we have “\(B\),” which is a set, and not a logical statement. It should be written as “\(x\in A\,\wedge\,x\in B \Rightarrow x\in A\cap B\).”

Exercise \(\PageIndex{14}\label{ex:unionint-14}\)

Prove or disprove each of the following statements about arbitrary sets \(A\) and \(B\). If you think a statement is true, prove it; if you think it is false, provide a counterexample.

(a) \(\mathscr{P}(A\cap B) = \mathscr{P}(A)\cap\mathscr{P}(B)\)

(b) \(\mathscr{P}(A\cup B) = \mathscr{P}(A)\cup\mathscr{P}(B)\)

(c) \(\mathscr{P}(A - B) = \mathscr{P}(A) - \mathscr{P}(B)\)

To show that two sets \(U\) and \(V\) are equal, we usually want to prove that \(U \subseteq V\) and \(V \subseteq U\).  For the subset relationship, we start with let \(x\in U \). In this problem, the element \(x\) is actually a set. Since we usually use uppercase letters to denote sets, for (a) we should start the proof of the subset relationship  “Let \(S\in\mathscr{P}(A\cap B)\),”  using an uppercase letter to emphasize the elements of \(\mathscr{P}(A\cap B)\) are sets. These remarks also apply to (b) and (c).

Exercise \(\PageIndex{15}\)

Let \({\cal U}=\{1,2,3,4,5,6,7,8\}\), \(A=\{2,4,6,8\}\), \(B=\{3,5\}\), \(C=\{1,2,3,4\}\) and\(D=\{6,8\}\). Find

(a) \(A\cap C\)                   (b) \(A\cap B\)                    (c) \(\emptyset \cup B\)   

(d) \(\emptyset \cap B\)                  (e) \(A-(B \cup C)\)              (f) \(C-B\)                   

(g) \(A\bigtriangleup C\)                (h) \(A \cup {\cal U}\)                  (i)  \(A\cap D\)                       

(j) \(A\cup D\)                  (k) \(B\cap D\)                      (l) \(B\bigtriangleup C\)   

(m)  \(A \cap {\cal U}\)                 (n) \(\overline{A}\)                      (o) \(\overline{B}\).                 

(p) \(D \cup (B \cap C)\)        (q) \(\overline{A \cup C}\)            (r) \(\overline{A} \cup \overline{C} \)

(s) Which pairs of sets are disjoint?

(a) \(\{2,4\}\)            (b) \(\emptyset \)       (c) \(B\)               (d)  \(\emptyset\)    

Exercise \(\PageIndex{16}\)

If \(A \subseteq B\) then \(A-B= \emptyset.\)

1.5 Set Operations with Three Sets

Learning objectives.

After completing this section, you should be able to:

• Interpret Venn diagrams with three sets.
• Create Venn diagrams with three sets.
• Apply set operations to three sets.
• Prove equality of sets using Venn diagrams.

Have you ever searched for something on the Internet and then soon after started seeing multiple advertisements for that item while browsing other web pages? Large corporations have built their business on data collection and analysis. As we start working with larger data sets, the analysis becomes more complex. In this section, we will extend our knowledge of set relationships by including a third set.

A Venn diagram with two intersecting sets breaks up the universal set into four regions; simply adding one additional set will increase the number of regions to eight, doubling the complexity of the problem.

Venn Diagrams with Three Sets

Below is a Venn diagram with two intersecting sets, which breaks the universal set up into four distinct regions.

Next, we see a Venn diagram with three intersecting sets , which breaks up the universal set into eight distinct regions.

Shading Venn Diagrams

Venn Diagram is an Android application that allows you to visualize how the sets are related in a Venn diagram by entering expressions and displaying the resulting Venn diagram of the set shaded in gray.

The Venn Diagram application uses some notation that differs from the notation covered in this text.

• The complement of set A A in this text is written symbolically as A ′ A ′ , but the Venn Diagram app uses A C A C to represent the complement operation.
• The set difference operation, − − , is available in the Venn Diagram app, although this operation is not covered in the text.

It is recommended that you explore this application to expand your knowledge of Venn diagrams prior to continuing with the next example.

In the next example, we will explore the three main blood factors, A, B and Rh. The following background information about blood types will help explain the relationships between the sets of blood factors. If an individual has blood factor A or B, those will be included in their blood type. The Rh factor is indicated with a + + or a − − . For example, if a person has all three blood factors, then their blood type would be AB + AB + . In the Venn diagram, they would be in the intersection of all three sets, A ∩ B ∩ R h + . A ∩ B ∩ R h + . If a person did not have any of these three blood factors, then their blood type would be O − , O − , and they would be in the set ( A ∪ B ∪ R h + ) ′ ( A ∪ B ∪ R h + ) ′ which is the region outside all three circles.

Example 1.35

Interpreting a venn diagram with three sets.

Use the Venn diagram below, which shows the blood types of 100 people who donated blood at a local clinic, to answer the following questions.

• How many people with a type A blood factor donated blood?
• Julio has blood type B + . B + . If he needs to have surgery that requires a blood transfusion, he can accept blood from anyone who does not have a type A blood factor. How many people donated blood that Julio can accept?
• How many people who donated blood do not have the Rh + Rh + blood factor?
• How many people had type A and type B blood?
• The number of people who donated blood with a type A blood factor will include the sum of all the values included in the A circle. It will be the union of sets A − , A + , A B − and A B + . A − , A + , A B − and A B + . n ( A ) = n ( A − ) + n ( A + ) + n ( A B − ) + n ( A B + ) = 6 + 36 + 1 + 3 = 46. n ( A ) = n ( A − ) + n ( A + ) + n ( A B − ) + n ( A B + ) = 6 + 36 + 1 + 3 = 46.
• In part 1, it was determined that the number of donors with a type A blood factor is 46. To determine the number of people who did not have a type A blood factor, use the following property, A ′ A ′ union is equal to U U , which means n ( A ) + n ( A ′ ) = n ( U ) , n ( A ) + n ( A ′ ) = n ( U ) , and n ( A ′ ) = n ( U ) − n ( A ) = 100 − 46 = 54. n ( A ′ ) = n ( U ) − n ( A ) = 100 − 46 = 54. Thus, 54 people donated blood that Julio can accept.
• This would be everyone outside the Rh + Rh + circle, or everyone with a negative Rh factor, n ( R h − ) = n ( O − ) + n ( A − ) + n ( A B − ) + n ( B − ) = 7 + 6 + 1 + 2 = 16. n ( R h − ) = n ( O − ) + n ( A − ) + n ( A B − ) + n ( B − ) = 7 + 6 + 1 + 2 = 16.
• To have both blood type A and blood type B, a person would need to be in the intersection of sets A A and B B . The two circles overlap in the regions labeled A B − A B − and A B + . A B + . Add up the number of people in these two regions to get the total: 1 + 3 = 4. 1 + 3 = 4. This can be written symbolically as n ( A and B ) = n ( A ∩ B ) = n ( A B − ) + n ( A B + ) = 1 + 3 = 4. n ( A and B ) = n ( A ∩ B ) = n ( A B − ) + n ( A B + ) = 1 + 3 = 4.

Your Turn 1.35

Blood types.

Most people know their main blood type of A, B, AB, or O and whether they are R h + R h + or R h − R h − , but did you know that the International Society of Blood Transfusion recognizes twenty-eight additional blood types that have important implications for organ transplants and successful pregnancy? For more information, check out this article:

Blood mystery solved: Two new blood types identified

Creating Venn Diagrams with Three Sets

In general, when creating Venn diagrams from data involving three subsets of a universal set, the strategy is to work from the inside out. Start with the intersection of the three sets, then address the regions that involve the intersection of two sets. Next, complete the regions that involve a single set, and finally address the region in the universal set that does not intersect with any of the three sets. This method can be extended to any number of sets. The key is to start with the region involving the most overlap, working your way from the center out.

Example 1.36

Creating a venn diagram with three sets.

A teacher surveyed her class of 43 students to find out how they prepared for their last test. She found that 24 students made flash cards, 14 studied their notes, and 27 completed the review assignment. Of the entire class of 43 students, 12 completed the review and made flash cards, nine completed the review and studied their notes, and seven made flash cards and studied their notes, while only five students completed all three of these tasks. The remaining students did not do any of these tasks. Create a Venn diagram with subsets labeled: “Notes,” “Flash Cards,” and “Review” to represent how the students prepared for the test.

Step 1: First, draw a Venn diagram with three intersecting circles to represent the three intersecting sets: Notes, Flash Cards, and Review. Label the universal set with the cardinality of the class.

Step 2: Next, in the region where all three sets intersect, enter the number of students who completed all three tasks.

Step 3: Next, calculate the value and label the three sections where just two sets overlap.

• Review and flash card overlap . A total of 12 students completed the review and made flash cards, but five of these twelve students did all three tasks, so we need to subtract: 12 − 5 = 7 12 − 5 = 7 . This is the value for the region where the flash card set intersects with the review set.
• Review and notes overlap . A total of 9 students completed the review and studied their notes, but again, five of these nine students completed all three tasks. So, we subtract: 9 − 5 = 4 9 − 5 = 4 . This is the value for the region where the review set intersects with the notes set.
• Flash card and notes overlap . A total of 7 students made flash cards and studied their notes; subtracting the five students that did all three tasks from this number leaves 2 students who only studied their notes and made flash cards. Add these values to the Venn diagram.

Step 4: Now, repeat this process to find the number of students who only completed one of these three tasks.

• A total of 24 students completed flash cards, but we have already accounted for 2 + 5 + 7 = 14 2 + 5 + 7 = 14 of these. Thus, 24 - 14 = 10 24 - 14 = 10 students who just made flash cards.
• A total of 14 students studied their notes, but we have already accounted for 4 + 5 + 2 = 11 4 + 5 + 2 = 11 of these. Thus, 14 - 11 = 3 14 - 11 = 3 students only studied their notes.
• A total of 27 students completed the review assignment, but we have already accounted for 4 + 5 + 7 = 16 4 + 5 + 7 = 16 of these, which means 27 - 16 = 11 27 - 16 = 11 students only completed the review assignment.
• Add these values to the Venn diagram.

Step 5: Finally, compute how many students did not do any of these three tasks. To do this, we add together each value that we have already calculated for the separate and intersecting sections of our three sets: 3 + 2 + 4 + 5 + 10 + 7 + 11 = 42 3 + 2 + 4 + 5 + 10 + 7 + 11 = 42 . Because there 43 students in the class, and 43 − 42 = 1 43 − 42 = 1 , this means only one student did not complete any of these tasks to prepare for the test. Record this value somewhere in the rectangle, but outside of all the circles, to complete the Venn diagram.

Your Turn 1.36

Applying set operations to three sets.

Set operations are applied between two sets at a time. Parentheses indicate which operation should be performed first. As with numbers, the inner most parentheses are applied first. Next, find the complement of any sets, then perform any union or intersections that remain.

Example 1.37

Perform the set operations as indicated on the following sets: U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } , A = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } , A = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } , B = { 0 , 2 , 4 , 6 , 8 , 10 , 12 } , B = { 0 , 2 , 4 , 6 , 8 , 10 , 12 } , and C = { 0 , 3 , 6 , 9 , 12 } . C = { 0 , 3 , 6 , 9 , 12 } .

• Find ( A ∩ B ) ∩ C . ( A ∩ B ) ∩ C .
• Find A ∩ ( B ∪ C ) . A ∩ ( B ∪ C ) .
• Find ( A ∩ B ) ∪ C ′ . ( A ∩ B ) ∪ C ′ .
• Parentheses first, A A intersection B B equals A ∩ B = { 0 , 2 , 4 , 6 } , A ∩ B = { 0 , 2 , 4 , 6 } , the elements common to both A A and B B . ( A ∩ B ) ∩ C = { 0 , 2 , 4 , 6 } ∩ { 0 , 3 , 6 , 9 , 12 } = { 0 , 6 } , ( A ∩ B ) ∩ C = { 0 , 2 , 4 , 6 } ∩ { 0 , 3 , 6 , 9 , 12 } = { 0 , 6 } , because the only elements that are in both sets are 0 and 6.
• Parentheses first, B B union C C equals B ∪ C = { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } , B ∪ C = { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } , the collection of all elements in set B B or set C C or both. A ∩ ( B ∪ C ) = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ∩ { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } = { 0 , 2 , 3 , 4 , 6 } , A ∩ ( B ∪ C ) = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ∩ { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } = { 0 , 2 , 3 , 4 , 6 } , because the intersection of these two sets is the set of elements that are common to both sets.
• Parentheses first, A A intersection B B equals A ∩ B = { 0 , 2 , 4 , 6 } . A ∩ B = { 0 , 2 , 4 , 6 } . Next, find C ′ . C ′ . The complement of set C C is the set of elements in the universal set U U that are not in set C . C . C ′ = { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } . C ′ = { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } . Finally, find ( A ∩ B ) ∪ C ′ = { 0 , 2 , 4 , 6 } ∪ { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } = { 0 , 1 , 2 , 4 , 5 , 6 , 7 , 8 , 10 , 11 } . ( A ∩ B ) ∪ C ′ = { 0 , 2 , 4 , 6 } ∪ { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } = { 0 , 1 , 2 , 4 , 5 , 6 , 7 , 8 , 10 , 11 } .

Your Turn 1.37

Using the same sets from Example 1.37, perform the set operations indicated.

Notice that the answers to the Your Turn are the same as those in the Example. This is not a coincidence. The following equivalences hold true for sets:

• A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C and A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C . A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C . These are the associative property for set intersection and set union.
• A ∩ B = B ∩ A A ∩ B = B ∩ A and A ∪ B = B ∪ A . A ∪ B = B ∪ A . These are the commutative property for set intersection and set union.
• A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) and A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . These are the distributive property for sets over union and intersection, respectively.

Proving Equality of Sets Using Venn Diagrams

To prove set equality using Venn diagrams, the strategy is to draw a Venn diagram to represent each side of the equality, then look at the resulting diagrams to see if the regions under consideration are identical.

Augustus De Morgan was an English mathematician known for his contributions to set theory and logic. De Morgan’s law for set complement over union states that ( A ∪ B ) ′ = A ′ ∩ B ′ ( A ∪ B ) ′ = A ′ ∩ B ′ . In the next example, we will use Venn diagrams to prove De Morgan’s law for set complement over union is true. But before we begin, let us confirm De Morgan’s law works for a specific example. While showing something is true for one specific example is not a proof, it will provide us with some reason to believe that it may be true for all cases.

Let U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , A = { 2 , 3 , 4 } , A = { 2 , 3 , 4 } , and B = { 3 , 4 , 5 , 6 } . B = { 3 , 4 , 5 , 6 } . We will use these sets in the equation ( A ∪ B ) ′ = A ′ ∩ B ′ . ( A ∪ B ) ′ = A ′ ∩ B ′ . To begin, find the value of the set defined by each side of the equation.

Step 1: A ∪ B A ∪ B is the collection of all unique elements in set A A or set B B or both. A ∪ B = { 2 , 3 , 4 , 5 , 6 } . A ∪ B = { 2 , 3 , 4 , 5 , 6 } . The complement of A union B , ( A ∪ B ) ′ ( A ∪ B ) ′ , is the set of all elements in the universal set that are not in A ∪ B A ∪ B . So, the left side the equation ( A ∪ B ) ′ ( A ∪ B ) ′ is equal to the set { 1 , 7 } . { 1 , 7 } .

Step 2: The right side of the equation is A ′ ∩ B ′ . A ′ ∩ B ′ . A ′ A ′ is the set of all members of the universal set U U that are not in set A A . A ′ = { 1 , 5 , 6 , 7 } . A ′ = { 1 , 5 , 6 , 7 } . Similarly, B ′ = { 1 , 2 , 7 } . B ′ = { 1 , 2 , 7 } .

Step 3: Finally, A ′ ∩ B ′ A ′ ∩ B ′ is the set of all elements that are in both A ′ A ′ and B ′ . B ′ . The numbers 1 and 7 are common to both sets, therefore, A ′ ∩ B ′ = { 1 , 7 } . A ′ ∩ B ′ = { 1 , 7 } . Because, { 1 , 7 } = { 1 , 7 } { 1 , 7 } = { 1 , 7 } we have demonstrated that De Morgan’s law for set complement over union works for this particular example. The Venn diagram below depicts this relationship.

Example 1.38

Proving de morgan’s law for set complement over union using a venn diagram.

De Morgan’s Law for the complement of the union of two sets A A and B B states that: ( A ∪ B ) ′ = A ′ ∩ B ′ . ( A ∪ B ) ′ = A ′ ∩ B ′ . Use a Venn diagram to prove that De Morgan’s Law is true.

Step 1: First, draw a Venn diagram representing the left side of the equality. The regions of interest are shaded to highlight the sets of interest. A ∪ B A ∪ B is shaded on the left, and ( A ∪ B ) ′ ( A ∪ B ) ′ is shaded on the right.

Step 2: Next, draw a Venn diagram to represent the right side of the equation. A ′ A ′ is shaded and B ′ B ′ is shaded. Because A ′ A ′ and B ′ B ′ mix to form A ′ ∩ B ′ A ′ ∩ B ′ is also shaded.

Step 3: Verify the conclusion. Because the shaded region in the Venn diagram for ( A ∪ B ) ′ ( A ∪ B ) ′ matches the shaded region in the Venn diagram for A ′ ∩ B ' A ′ ∩ B ' , the two sides of the equation are equal, and the statement is true. This completes the proof that De Morgan’s law is valid.

Your Turn 1.38

Check your understanding, section 1.5 exercises.

A gamers club at Baily Middle School consisting of 25 members was surveyed to find out who played board games, card games, or video games. Use the results depicted in the Venn diagram below to answer the following exercises.

A blood drive at City Honors High School recently collected blood from 140 students, staff, and faculty. Use the results depicted in the Venn diagram below to answer the following exercises.

For the following exercises, perform the set operations as indicated on the following sets: U = { 20 , 21 , 22 , … , 29 } , A = { 21 , 24 , 27 } , B = { 20 , 22 , 24 , 28 } , and C = { 21 , 23 , 25 , 27 } .

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Sets - Problem Solving

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If \(A = \{ 1,2,3,4 \}, B = \{ 4,5,6,7 \},\) determine the following sets: (i) \(A \cap B\) (ii) \(A \cup B\) (iii) \(A \backslash B \) (i) By definition, \(\cap\) tells us that we want to find the common elements between the two sets. In this case, it is 4 only. Thus \(A \cap B = \{ 4 \} \). (ii) By definition, \(\cup\) tells us that we want to combine all the elements between the two sets. In this case, it is \(A\cup B = \{1,2,3,4,5,6,7 \} \). (iii) By definition, \( \backslash \) tells us that we want to look for elements in the former set in that doesn't appear in the latter set. So \(A \backslash B = \{1,2,3\} \). \(_\square\)

Consider the same example above. If the element \(4\) is removed from the set \(B\), solve for (i), (ii), (iii) as well. (i) Since there is no common elements in sets \(A\) and \(B\), then \(A \cap B = \phi \) or \(A \cap B = \{ \} \). (ii) Because the element \(4\) is no longer repeated, then \(A \cup B \) remains the same. (iii) Since \(A\) and \(B\) no longer share any common element, \(A\backslash B \) is simply equals to set \(A\), which is \(\{1,2,3,4 \} \). \(_\square\)

If \(P=\{2, 5, 6, 3, 7\}\) and \(Q=\{1, 2, 3, 8, 9, 10\},\) which of the following Venn diagrams represents the relationship between the two sets?

\[\large\color{darkred}{B=\{ \{ M,A,T,H,S \} \}}\]

Find the cardinal number of the set \(\color{darkred}{B}\).

Note: The cardinal number of a set is equal to the number of elements contained in the set.

Bonus question given with the picture.

Join the brilliant classes and enjoy the excellence. also checkout foundation assignment #2 for jee..

Consider the set \( \lbrace{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\rbrace}\).

For each of its subsets, let \( M \) be the greatest number. Find the last three digits of the sum of all the \( M \)'s.

Assume that \(0\) is the greatest number of the empty subset.

The number of subsets in set A is 192 more than the number of subsets in set B. How many elements are there in set A?

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Statistics and probability

Course: statistics and probability   >   unit 7.

• Intersection and union of sets
• Relative complement or difference between sets
• Universal set and absolute complement
• Subset, strict subset, and superset
• Bringing the set operations together

Basic set notation

• (Choice A)   { 5 , 10 , 15 , 20 , 30 } ‍   A { 5 , 10 , 15 , 20 , 30 } ‍  
• (Choice B)   { 5 , 15 , 30 } ‍   B { 5 , 15 , 30 } ‍  
• (Choice C)   { 10 , 20 } ‍   C { 10 , 20 } ‍  
• (Choice D)   { } ‍   D { } ‍  

Word Problems on Sets

Word problems on sets are solved here to get the basic ideas how to use the  properties of union and intersection of sets.

Solved basic word problems on sets:

1. Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩ B).

Solution:  Using the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B).  then n(A ∩ B) = n(A) + n(B) - n(A ∪ B)                       = 20 + 28 - 36                       = 48 - 36                       = 12 

2.  If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B).

Solution:  Using the formula n(A∪B) = n(A - B) + n(A ∩ B) + n(B - A)                                   70 = 18 + 25 + n(B - A)                                   70 = 43 + n(B - A)                           n(B - A) = 70 - 43                           n(B - A) = 27  Now n(B) = n(A ∩ B) + n(B - A)                 = 25 + 27                 = 52 

Different types on word problems on sets:

3.  In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both coffee and tea? 

Solution:  Let A = Set of people who like cold drinks.       B = Set of people who like hot drinks.  Given   (A ∪ B) = 60            n(A) = 27       n(B) = 42 then;

n(A ∩ B) = n(A) + n(B) - n(A ∪ B)              = 27 + 42 - 60              = 69 - 60 = 9              = 9  Therefore, 9 people like both tea and coffee. 

4.  There are 35 students in art class and 57 students in dance class. Find the number of students who are either in art class or in dance class.

•  When two classes meet at different hours and 12 students are enrolled in both activities.  •  When two classes meet at the same hour.  Solution:  n(A) = 35,       n(B) = 57,       n(A ∩ B) = 12  (Let A be the set of students in art class.  B be the set of students in dance class.)  (i) When 2 classes meet at different hours n(A ∪ B) = n(A) + n(B) - n(A ∩ B)                                                                            = 35 + 57 - 12                                                                            = 92 - 12                                                                            = 80  (ii) When two classes meet at the same hour, A∩B = ∅ n (A ∪ B) = n(A) + n(B) - n(A ∩ B)                                                                                                = n(A) + n(B)                                                                                                = 35 + 57                                                                                                = 92

Further concept to solve word problems on sets:

5. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?

Solution: Let A be the set of people who speak English. B be the set of people who speak French. A - B be the set of people who speak English and not French. B - A be the set of people who speak French and not English. A ∩ B be the set of people who speak both French and English. Given, n(A) = 72       n(B) = 43       n(A ∪ B) = 100 Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)                      = 72 + 43 - 100                      = 115 - 100                      = 15 Therefore, Number of persons who speak both French and English = 15 n(A) = n(A - B) + n(A ∩ B) ⇒ n(A - B) = n(A) - n(A ∩ B)                 = 72 - 15                 = 57 and n(B - A) = n(B) - n(A ∩ B)                    = 43 - 15                    = 28 Therefore, Number of people speaking English only = 57 Number of people speaking French only = 28

Word problems on sets using the different properties (Union & Intersection):

6. In a competition, a school awarded medals in different categories. 36 medals in dance, 12 medals in dramatics and 18 medals in music. If these medals went to a total of 45 persons and only 4 persons got medals in all the three categories, how many received medals in exactly two of these categories?

Solution: Let A = set of persons who got medals in dance. B = set of persons who got medals in dramatics. C = set of persons who got medals in music. Given, n(A) = 36                              n(B) = 12       n(C) = 18 n(A ∪ B ∪ C) = 45       n(A ∩ B ∩ C) = 4 We know that number of elements belonging to exactly two of the three sets A, B, C = n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3n(A ∩ B ∩ C) = n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3 × 4       ……..(i) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C) Therefore, n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) From (i) required number = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) - 12 = 36 + 12 + 18 + 4 - 45 - 12 = 70 - 57 = 13

Apply set operations to solve the word problems on sets:

7. Each student in a class of 40 plays at least one indoor game chess, carrom and scrabble. 18 play chess, 20 play scrabble and 27 play carrom. 7 play chess and scrabble, 12 play scrabble and carrom and 4 play chess, carrom and scrabble. Find the number of students who play (i) chess and carrom. (ii) chess, carrom but not scrabble.

Solution: Let A be the set of students who play chess B be the set of students who play scrabble C be the set of students who play carrom Therefore, We are given n(A ∪ B ∪ C) = 40, n(A) = 18,         n(B) = 20         n(C) = 27, n(A ∩ B) = 7,     n(C ∩ B) = 12    n(A ∩ B ∩ C) = 4 We have n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C) Therefore, 40 = 18 + 20 + 27 - 7 - 12 - n(C ∩ A) + 4 40 = 69 – 19 - n(C ∩ A) 40 = 50 - n(C ∩ A) n(C ∩ A) = 50 - 40 n(C ∩ A) = 10 Therefore, Number of students who play chess and carrom are 10. Also, number of students who play chess, carrom and not scrabble. = n(C ∩ A) - n(A ∩ B ∩ C) = 10 – 4 = 6

Therefore, we learned how to solve different types of word problems on sets without using Venn diagram.

● Set Theory

● Sets Theory

● Representation of a Set

● Types of Sets

● Finite Sets and Infinite Sets

● Power Set

● Problems on Union of Sets

● Problems on Intersection of Sets

● Difference of two Sets

● Complement of a Set

● Problems on Complement of a Set

● Problems on Operation on Sets

● Word Problems on Sets

● Venn Diagrams in Different Situations

● Relationship in Sets using Venn Diagram

● Union of Sets using Venn Diagram

● Intersection of Sets using Venn Diagram

● Disjoint of Sets using Venn Diagram

● Difference of Sets using Venn Diagram

● Examples on Venn Diagram

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Set covering problem

Authors: Sherry Liang, Khalid Alanazi, Kumail Al Hamoud (ChemE 6800 Fall 2020)

• 1 Introduction
• 2 Problem formulation
• 3 Integer linear program formulation
• 4 Approximation via LP relaxation and rounding
• 5 Greedy approximation algorithm
• 6 Numerical Example
• 7 Applications
• 8 Conclusion
• 9 References

Introduction

The set covering problem is a significant NP-hard problem in combinatorial optimization. Given a collection of elements, the set covering problem aims to find the minimum number of sets that incorporate (cover) all of these elements. [1]

The set covering problem importance has two main aspects: one is pedagogical, and the other is practical.

First, because many greedy approximation methods have been proposed for this combinatorial problem, studying it gives insight into the use of approximation algorithms in solving NP-hard problems. Thus, it is a primal example in teaching computational algorithms. We present a preview of these methods in a later section, and we refer the interested reader to these references for a deeper discussion. [1] [2] [3]

Second, many problems in different industries can be formulated as set covering problems. For example, scheduling machines to perform certain jobs can be thought of as covering the jobs. Picking the optimal location for a cell tower so that it covers the maximum number of customers is another set covering application. Moreover, this problem has many applications in the airline industry, and it was explored on an industrial scale as early as the 1970s. [4]

Problem formulation

Integer linear program formulation

An integer linear program (ILP) model can be formulated for the minimum set covering problem as follows: [1]

Decision variables

Objective function

Constraints

Set covering problems are significant NP-hard optimization problems, which implies that as the size of the problem increases, the computational time to solve it increases exponentially. Therefore, there exist approximation algorithms that can solve large scale problems in polynomial time with optimal or near-optimal solutions. In subsequent sections, we will cover two of the most widely used approximation methods to solve set cover problem in polynomial time which are linear program relaxation methods and classical greedy algorithms. [2]

Approximation via LP relaxation and rounding

The above LP formulation is a relaxation of the original ILP set cover problem. This means that every feasible solution of the integer program is also feasible for this LP program. Additionally, the value of any feasible solution for the integer program is the same value in LP since the objective functions of both integer and linear programs are the same. Solving the LP program will result in an optimal solution that is a lower bound for the original integer program since the minimization of LP finds a feasible solution of lowest possible values. Moreover, we use LP rounding algorithms to directly round the fractional LP solution to an integral combinatorial solution as follows:

Deterministic rounding algorithm

Greedy approximation algorithm

Greedy algorithm for minimum set cover example:

Numerical Example

Let’s consider a simple example where we assign cameras at different locations. Each location covers some areas of stadiums, and our goal is to put the least amount of cameras such that all areas of stadiums are covered. We have stadium areas from 1 to 15, and possible camera locations from 1 to 8.

We are given that camera location 1 covers stadium areas {1,3,4,6,7}, camera location 2 covers stadium areas {4,7,8,12}, while the remaining camera locations and the stadium areas that the cameras can cover are given in table 1 below:

s.t. Constraints 1 to 15 are satisfied:

To conclude, our two solutions are:

The minimum number of cameras that we need to install is 4.

Let's now consider solving the problem using the greedy algorithm.

First Iteration:

Second Iteration:

Third Iteration:

Fourth Iteration:

The solution we obtained is:

The greedy algorithm does not provide the optimal solution in this case.

The usual elimination algorithm would give us the minimum number of cameras that we need to install to be4, but the greedy algorithm gives us the minimum number of cameras that we need to install is 5.

Applications

The applications of the set covering problem span a wide range of applications, but its usefulness is evident in industrial and governmental planning. Variations of the set covering problem that are of practical significance include the following.

This set covering problems is concerned with maximizing the coverage of some public facilities placed at different locations. [7] Consider the problem of placing fire stations to serve the towns of some city. [8] If each fire station can serve its town and all adjacent towns, we can formulate a set covering problem where each subset consists of a set of adjacent towns. The problem is then solved to minimize the required number of fire stations to serve the whole city.

A real-world case study involving optimizing fire station locations in Istanbul is analyzed in this reference. [8] The Istanbul municipality serves 790 subdistricts, which should all be covered by a fire station. Each subdistrict is considered covered if it has a neighboring district (a district at most 5 minutes away) that has a fire station. For detailed computational analysis, we refer the reader to the mentioned academic paper.

The work by Ali and Dyo explores a greedy approximation algorithm to solve an optimal selection problem including 713 bus routes in Greater London. [9] Using 14% of the routes only (100 routes), the greedy algorithm returns a solution that covers 25% of the segments in Greater London. For a details of the approximation algorithm and the world case study, we refer the reader to this reference. [9] For a significantly larger case study involving 5747 buses covering 5060km, we refer the reader to this academic article. [10]

An important application of large-scale set covering is the airline crew scheduling problem, which pertains to assigning airline staff to work shifts. [4] [11] Thinking of the collection of flights as a universal set to be covered, we can formulate a set covering problem to search for the optimal assignment of employees to flights. Due to the complexity of airline schedules, this problem is usually divided into two subproblems: crew pairing and crew assignment. We refer the interested reader to this survey, which contains several problem instances with the number of flights ranging from 1013 to 7765 flights, for a detailed analysis of the formulation and algorithms that pertain to this significant application. [4] [12]

The set covering problem, which aims to find the least number of subsets that cover some universal set, is a widely known NP-hard combinatorial problem. Due to its applicability to route planning and airline crew scheduling, several methods have been proposed to solve it. Its straightforward formulation allows for the use of off-the-shelf optimizers to solve it. Moreover, heuristic techniques and greedy algorithms can be used to solve large-scale set covering problems for industrial applications.

• ↑ 1.0 1.1 1.2 1.3 1.4 T. Grossman and A. Wool, "Computational experience with approximation algorithms for the set covering problem ," European Journal of Operational Research , vol. 101, pp. 81-92, 1997.
• ↑ 2.0 2.1 2.2 P. Slavı́k, "A Tight Analysis of the Greedy Algorithm for Set Cover ," Journal of Algorithms, , vol. 25, pp. 237-245, 1997.
• ↑ 3.0 3.1 T. Grossman and A. Wool, "What Is the Best Greedy-like Heuristic for the Weighted Set Covering Problem? ," Operations Research Letters , vol. 44, pp. 366-369, 2016.
• ↑ 4.0 4.1 4.2 J. Rubin, "A Technique for the Solution of Massive Set Covering Problems, with Application to Airline Crew Scheduling ," Transportation Science , vol. 7, pp. 34-48, 1973.
• ↑ 5.0 5.1 5.2 Williamson, David P., and David B. Shmoys. “The Design of Approximation Algorithms” [1] . “Cambridge University Press”, 2011.
• ↑ V. Chvatal, "Greedy Heuristic for the Set-Covering Problem ," Mathematics of Operations Research , vol. 4, pp. 233-235, 1979.
• ↑ R. Church and C. ReVelle, "The maximal covering location problem ," Papers of the Regional Science Association , vol. 32, pp. 101-118, 1974.
• ↑ 8.0 8.1 E. Aktaş, Ö. Özaydın, B. Bozkaya, F. Ülengin, and Ş. Önsel, "Optimizing Fire Station Locations for the Istanbul Metropolitan Municipality ," Interfaces , vol. 43, pp. 240-255, 2013.
• ↑ 9.0 9.1 9.2 J. Ali and V. Dyo, "Coverage and Mobile Sensor Placement for Vehicles on Predetermined Routes: A Greedy Heuristic Approach ," Proceedings of the 14th International Joint Conference on E-Business and Telecommunications , pp. 83-88, 2017.
• ↑ 10.0 10.1 P.H. Cruz Caminha , R. De Souza Couto , L.H. Maciel Kosmalski Costa , A. Fladenmuller , and M. Dias de Amorim, "On the Coverage of Bus-Based Mobile Sensing ," Sensors , 2018.
• ↑ E. Marchiori and A. Steenbeek, "An Evolutionary Algorithm for Large Scale Set Covering Problems with Application to Airline Crew Scheduling ," Real-World Applications of Evolutionary Computing. EvoWorkshops 2000. Lecture Notes in Computer Science , 2000.
• ↑ A. Kasirzadeh, M. Saddoune, and F. Soumis "Airline crew scheduling: models, algorithms, and data sets ," EURO Journal on Transportation and Logistics , vol. 6, pp. 111-137, 2017.

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Commentary | The Key Bridge disaster is a wake-up call for…

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Commentary | The Key Bridge disaster is a wake-up call for US infrastructure | GUEST COMMENTARY

President Biden visited the site of the collapsed Francis Scott Key Bridge in Baltimore, which has been called a “ mangled mess ,” on Friday, mourning the loss of the six workers killed in the accident. The economic impact of this accident is significant and goes far beyond the bridge itself . The disaster should serve as a wake-up call to ensure that all bridges, as well as all marine infrastructure in general, can withstand such a collision. Rebuilding the Key Bridge offers the chance to bring American infrastructure delivery up to rest-of-world standards through better cooperation between the public and private sectors.

The core of such cooperation is a long-term contract between the public and private sectors known broadly as a “public-private partnership” or PPP.  The success of public-private partnerships in building, operating and maintaining all areas of infrastructure can be seen all over the country; and the future of PPP starts with improving public policy, including assessing standards .

The central aspect of a PPP is that it bundles or “wraps” the design and construction of a piece of infrastructure together with its operation and maintenance over the long term, such as 25 or 30 years. A PPP might also include private-sector financing to cover the new bridge’s substantial design and construction costs. Because the Key Bridge featured all-electronic tolling, a user-fee funding source already exists to help pay for the new bridge over time. This means that the new bridge can be delivered quickly under a “toll concession” agreement.

The federal government  already committed some funding to reconstruct the Key Bridge. However, a new bridge will be expensive and will cost more than this initial commitment. A PPP for the Key Bridge would combine design and construction with operation and maintenance — and include provisions to ensure that the infrastructure is properly maintained. This would reduce the likelihood of deferred maintenance, one of the main problems plaguing U.S. infrastructure today.

Rather than simply bouncing back from this disaster, a long-term PPP provides the opportunity to “bounce forward.” Since the Key Bridge first entered into service in 1977, a quiet, but vast, technological revolution has occurred in infrastructure. Improvements in materials (such as concrete and asphalt), sensors, designs and much more are readily available. These improvements can be incorporated into the new bridge’s design and construction, as well as its operation and maintenance by “future-proofing” PPP contracts with the private sector.

PPP contracts include operation and maintenance over the long term, which risks locking in outdated technologies. Future-proofing refers to the risk of not adopting available innovative technology and design standards well into the future. A future-proofed contract thus ensures that private capital, incentives and expertise are deployed to make U.S. infrastructure as resilient as possible for decades to come.

Moreover, many U.S. infrastructure projects notoriously run over both time and budget. When completed, Phase 1 of New York’s Second Avenue Subway, for example, cost about $2.5 billion per mile . That is 8 to 12 times more expensive than similar subway projects in Sweden, Italy, Paris, Berlin and Istanbul. There is no need for the new Key Bridge to suffer the same fate.  A PPP contract puts the risk of time and cost overruns on the private partner rather than the taxpayer. The private partner can be incentivized to deliver the project on time via financial penalties for late delivery and rewards for delivery ahead of schedule. A PPP would deliver the new bridge faster.

PPPs also allow projects to cut through much of the bureaucracy that often slows U.S. projects. America has typically used a design-bid-build (DBB) approach, where a government entity first bids out the bridge’s design and then bids out the chosen design separately. Combining and integrating the design and construction into a single project allows for quicker delivery and more synergies between design and construction firms. New York’s widely acclaimed new Tappan Zee Bridge was built using such a contract.

The United States is  decades behind many other countries where the use of PPPs is standard. Despite the suffering caused by this tragic accident, it provides an opportunity for Maryland to “bounce forward” and prevent future degradation of current infrastructure systems.

R. Richard Geddes ([email protected]) is a Nonresident Senior Fellow at the American Enterprise Institute, a professor in Cornell’s Jeb E. Brooks School of Public Policy and the Founding Director of the Cornell Program in Infrastructure Policy. He is a 1984 graduate of Towson State University.

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