A Venn diagram is a diagrammatic representation of logical relationships between a finite number of sets. John Venn popularized this method in the \(1880{\rm{s}}\) to help illustrate the inclusion and exclusion relationships of sets. These are also called logic diagrams, set diagrams, or primary diagrams. These form a logical foundation for fields such as geometry and topology and are commonly employed in Statistics, Computer Science, and Engineering too!

Sets: Definition

Generally, the term set refers to a collection of objects. In mathematics, a set refers to a collection of elements. The elements defined in a set can be any mathematical objects such as numbers, points, lines, symbols, geometrical shapes, variables, functions, or even other smaller sets. The number of elements in a set may be countable or uncountable.  

Sets: Examples

1. The set of letters in the English alphabet is countable. The set will have \(26\) elements from \(A\) to \(Z\).
2. The set of real numbers is infinite.

Representing a Set

A set is represented as an uppercase letter, and it is written by listing all the members separated by commas and enclosed in braces.

Cardinality of a Set

Cardinality of a set is the measure of the size of a set. It represents the number of elements in a set. Cardinality is represented by vertical bars. For example, for a set \(A\), cardinality is written as \(\left| A \right|\). It is also represented as \(n(A)\).

Types of Sets

1. A set with no elements is called an empty set or a null set. A null set is denoted by \(\phi \).
2. A set with only one element is called a singleton set.
3. A set may have an infinite number of elements. As it is impossible to list all the elements in an infinite set, a formula usually represents it.
4. A universal set is a set that has all the elements of all the related sets. The elements in a universal set are not repeated. It is denoted by \(U\).

Observe that the only condition for the elements to be in a set is that they must all be connected by the same rule of law. Also, this rule must be universal and not subjective. For example, I cannot make a set with tall people. Being tall is very subjective. People who are tall for me may not be considered tall for someone who is \(6\) feet tall themselves.

Drawing a Venn Diagram

A Venn diagram is drawn using simple curves and lines on a plane. They usually consist of overlapping circles. Any closed figure can be used in place of the circle. Circles are usually preferred as it is easy to draw overlapping circles. The inside of each circle represents the elements that are members of a set. The outside of the circle represents the elements that are not members of the set. The overlapping region will contain the elements that are members of all the overlapping sets.

Rules to Draw Venn Diagrams

1. Know your universal set. It is usually drawn as a quadrilateral. It is denoted by \(U\).
2. Every set to be drawn will be a subset of the universal set. They are drawn as circles. There may be any number of circles. For ease of use, it is generally restricted to \(3\) circles or subsets.
3. Each circle is labelled with the name of the set to help in identification.
4. The elements of each set are written in the corresponding circles.
5. The elements that are members of more than one set are written in the overlapping area.

A simple Venn diagram of two overlapping sets, \(A\) and \(B\), will appear as shown below.

Benefits of Venn Diagram

1. A Venn diagram can be used for both comparison and classification.
2. It categorizes information.
3. It helps students visualize the relationships between different sets.
4. It helps reason complex issues using logic.
5. Highlights data patterns that may not be visible otherwise.

Operations on Sets

Four basic operations are carried out on sets.

Operation	Notation	Meaning
Intersection	\(A \cap B\)	denotes elements that are members of both sets \(A\) and \(B\)
Union	\(A \cup B\)	denotes all the elements in sets \(A\) and \(B\)
Difference	\(A – B\)	denotes elements that are in set \(A\) but not in set \(B\)
Complement	\(\overline A \) or \(A’\)	denotes elements that are not members of set \(A\)

Venn Diagram Operations on Sets

Let us now try to combine the concept of Venn diagrams and operations using two sets, \(A\) and \(B\).

1. Intersection of Sets

The intersection of sets is nothing but elements that are common to sets \(A\) and \(B\). In a Venn diagram, this is represented in the overlapping region of the two circles. The intersection of two sets is denoted as \(A \cap B\).

Intersection of two sets can also be defined as

\(n(A \cap B) = n(A) + n(B) – n(A \cup B)\)

Here, \(n\left( A \right) \to \) cardinality of \(A\)

\(n\left(B \right) \to \) cardinality of \(B\)

2. Union of Sets

Union of sets is a set that contains all the elements that are present in either \(A\) or \(B\). In a Venn diagram, union is represented as the complete area of both circles. Union of two sets is denoted as \(A \cup B\).

Union of two sets can also be defined as

\(n(A \cup B) = n(A) + n(B) – n(A \cap B)\)

Here, \(n\left( A \right) \to \) cardinality of \(A\)

\(n\left(B \right) \to \) cardinality of \(B\)

3. Difference of Sets

The difference between sets \(A\) and \(B\) is a set that lists the elements in \(A\) but not in set \(B\). It is denoted as \(A – B\), read as \(A\) difference \(B\). This is also called relative complement.

In the figure, observe that \(A\) difference \(B\) is not the same as \(B\) difference \(A\). This inequality means that difference of sets is not commutative. This is written as:

\(A\, – \,B \ne B\, – \,A\)

Note that union and intersection of sets are commutative. That is,

1. \(A \cup B = B \cup A\)
2. \(A \cap B = B \cap A\)

4. Complement of a Set

The elements in a complement set \(A’\) (read as \(A\)-dash) are not present in set \(A\). It can also be denoted as \(\overline A \).

From the figure, observe that \(A + A’ = U.\)

This means that the set formed with elements in \(A\) and \(A’\) makes \(U\).

1. The complement of a complement of a set is the set itself. It can be written as \({\left( {{A^\prime }} \right)^\prime } = A\)
2. Complement of a set \(A\) can also be written as \({A^\prime } = U – A\).

Solved Examples – Venn Diagram Operations on Sets

Q.1. \(A = \left\{ { 1,\, 3,\, 7,\,9,} \right\},\,B = \left\{ { 3,\,7,\,8,\,10} \right\}\)
Find a) \(A \cup B\) b) \(A \cap B\)
Ans:

a. \(A \cup B = \left\{{ 1,\,3,\,7,\,8,\,9,\,10} \right\} \)
b. \(A \cap B = \left\{{ 3,\,7} \right\} \)

Q.2. \(P = \left\{ { – 6,\, – 4,\, – 0.5,\,0,\,1,\,6,\,8} \right\},\,Q = \left\{ { – 0.5,\,0,\,1,\,2,\,4} \right\}\)
Find a. \(P – Q\) b. \(Q – P\)
Ans:

a. \(P – Q = \left\{ { – 6,\, – 4,\,6,\,8} \right\}\)

b. \(Q – P = \left\{ {2,\,4} \right\}\)

Q.3. \(U = \left\{ {1,\,12,\,23,\,2,\,6,\,7,\,11,\,10,\,16} \right\}\)
\(A = \left\{ {1,\,2,\,5,\,7,\,8,\,9,\,10} \right\}\)
\(B = \left\{ {2,\,6,\,12,\,10,\,16} \right\}\)
Find a. \(A’\) b. \(B’\) c. \({(A \cup B)^\prime }\)
Ans:

a. \(A’ = \left\{ {2,\,10,\,6,\,12,\,16,\,23,\,11} \right\}\)

b. \(B’ = \left\{ {1,\,5,\,7,\,8,\,9,\,23,\,11} \right\}\)

c. \(\left( {A \cup B} \right)’ = \left\{ {23,\,11} \right\}\)

Q.4. At a breakfast buffet, \(93\) people chose to have coffee, and \(47\) people chose juice. \(25\) people chose both coffee and juice. If each person chose at least one of the beverages, how many people dined at the buffet?
Ans: \(n\left( {{\rm{coffee}} \cup {\rm{juice}}} \right) = n\left( {{\rm{coffee}}} \right) + n\left( {{\rm{juice}}} \right) – n\left( {{\rm{coffee}} \cap {\rm{juice}}} \right)\)
Here,
\(n{\rm{(coffee) }}\) → number of people who drank coffee \(=93\)
\(n{\rm{(juice) }}\) → number of people who drank juice \(=47\)
\(n({\rm{coffee}} \cap {\rm{juice}})\) → number of people who had both \(n\left( {{\rm{coffee}} \cup {\rm{juice}}} \right)\) → total number of people at the buffet
Therefore, \(n\left( {{\rm{coffee}} \cup {\rm{juice}}} \right) = 93 + 47 – 25\)
Total number of people at the buffet = \(115\)

Q.5. In a class of \(110\) students, \(35\) like Science and \(45\) like Math. \(10\) like both. How many like either of them, and how many like neither?
Ans:
Given:
Total number of students, \(n\left( U \right) = {\rm{100}}\)
Number of students who like Science, \(n\left( S \right) = {\rm{35}}\)
Number of students who like Math, \(n\left( M \right) = {\rm{45}}\)
Number of students who like both, \(n(S \cap M) = 10\)

Number of students who like either \( = n(S \cup M)\)
\(n(S \cup M) = n(S) + n(M) – n(S \cap M)\)
\(n(S \cup M) = 35 + 45 – 10\)
\(n(S \cup M) = 70\)
Number of students who like neither \( = n{(S \cup M)^\prime }\)
\(n{(S \cup M)^\prime } = n(U) – n{(S \cup M)^\prime }\)
\(n{(S \cup M)^\prime } = 100 – 70\)
\(n{(S \cup M)^\prime } = 30\)

Summary

This article covers the various operations performed on sets using Venn diagrams after explaining the definition and examples for sets in mathematics. It also explains how Venn diagrams were first used and the steps to draw one. The four main operations performed using Venn diagrams are – intersection, union, difference, and complement. While the first three operations are performed on two or more sets, the fourth operation is usually performed on a single set or a resulting set of the first three operations.

Learn About Representation of Sets

Frequently Asked Questions (FAQs)

Q.1. How do you use Venn diagrams to solve set operations?
Ans: In a Venn diagram, the universal set is represented by a quadrilateral, usually a rectangle. The sets are denoted by a circular or oval shape within this quadrilateral. The overlapping region of the circles denotes the intersection of the sets. This diagram can further be used to solve set operations.

Q.2. What are the operations between sets?
Ans: There are four basic operations between the sets. They are:
1. Union of sets
2. Intersection of sets
3. Difference of sets
4. Complement of a set

Q.3. What does AUB represent?
Ans: The union of \(A\) and \(B\) is represented as \(A \cup B\). It is the set that has all the elements present in \(A\) or \(B\). This operation is similar to the addition of two sets. Here is the Venn diagram that shows the union operation.

Q.4. What does A∩B mean in the Venn diagram?
Ans: The intersection of the two sets \(A\) and \(B\) is written as \(A \cap B\). This set has the elements present in \(A\) and \(B\). In a Venn diagram, the intersection is denoted by the area overlapped by the circles.

This cardinality of this operation can also be defined as:
\((A \cap B) = n(A) + n(B) – n(A \cup B)\)

Q.5. Why do we use the Venn diagram in illustrating sets?
Ans: Venn diagrams are easy means to organize data. They provide a ready visual of the logical relationships between the sets. They also highlight the similarities and differences of the sets used.

We hope this detailed article on Venn diagram operations on sets helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!