The Venn diagram also called as the set diagram is used to depict set operations. It was discovered by John Venn and hence named after him. It was also used by Leonard Euler, the great Swiss Mathematician. So, these are sometimes called as the Venn - Euler Diagram or simply the Venn diagram.

These diagrams show the logical relationship among the sets under consideration. These relationships include all three basic operations - Intersection, Union, Complement. The Venn Diagram makes the entire concept very clear. Venn diagrams are also used in other fields like Statistics, Computer Science etc.

What is a Venn Diagram?

The Venn diagram is a pictorial representation of the given data. Here, we shall represent different operations that can be operated under sets and represented by the venn diagram.

A set X is called a universal set, if every set under consideration is a subset of X. The universal set is not a fixed set. In Venn diagrams, the universal set is depicted by the interior of a rectangle, whereas the subsets of universal sets are represented by the interior of circles, ellipses etc.

A set is a collection of objects out of a universal set and the members of the set are the objects contained within the set.

Let us consider U as the universal set and A is any subset of the universal set. Then it is represented in the Venn-diagram as:

How to Make a Venn Diagram?

A Venn Diagram is constructed with a collection of closed curves mostly circles of the same size. These circles overlap each other. The different regions formed by these overlapping circles represent the various set operations. These regions are filled either by the elements of the sets or by the number of elements of the sets. Venn diagrams are very similar to the Euler Diagrams.

The following is the Venn Diagram for any two non-empty sets A and B:

Following is the Venn Diagram for three non-empty Sets A and B:

Some other important Venn Diagrams are as follows:

A ∩ B

The green shaded portion represents the elements belonging to the set A âˆ© B.

A ∪ B

The green shaded portion represents the elements belonging to the set A âˆª B.

(A ∪ B)'

The green shaded portion represents the elements belonging to the Set (A âˆ© B)'.

(A ∩ B)'

The green shaded portion represents the elements belonging to the set (A ∪ B)'.

Euler Diagrams

Euler diagrams, which is an effective and intuitive way of representing relationships between sets. Euler diagrams consist of closed curves which divides the plane into zones, thus it provides an effective and intuitive way for representing the relationship between sets.

A Euler diagram is show above. One of the common interpretation of a Euler diagram is a set of intersection. With this intersection, the area in the above diagram representing the sets is X, Y and Z. The diagram also includes the intersections, X&Y, X&Z and X&Y&Z. From the diagram, no area represents the set not X & Z, set Z completely exists in X.

Triple Venn Diagrams

We use the triple Venn Diagram to compare and contrast the sets. You can see the Triple Venn Diagram below.

Examples on Making Triple Venn Diagrams

Given below are some examples on making the triple Venn diagram.

Example 1:

Represent the following sets as a Venn diagram and then find A', B', A ∩ B ∩ C, A ∪ B, (A ∩ B ∩ C)', (A ∪ B)'.

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 4, 5, 7, 8}

B = {1, 2, 5, 9, 10}

C = {3, 4, 5, 9, 10}

Solution:

The Venn Diagram representing the sets U, A, B and C is shown as follows:

A' = {2, 3, 6, 9, 10}

B' = {3, 4, 6, 7, 8}

A ∩ B ∩ C = {1, 5}

A ∪ B = {1, 2, 4, 5, 7, 8, 9, 10}

(A ∩ B)' = {2, 3, 4, 6, 7, 8, 9, 10}

(A ∪ B ∪ C)' = {6}

Venn Diagram Problems

Given below are some example based on Venn diagram.

Example 1:

There are 50 players who participate in the tournament match. Among the 50 players, 25 player will play in base ball match and the remaining 25 players will play in a foot ball match and 10 players will play in both the base ball and foot ball match. Solve this problem by using the Venn diagram. Calculate how many players will play in match and how many wont play in the match?

Solution:

There are two categories, one is base ball and another one is foot ball.

Step 1: We shall draw the diagram depending on the classification of the given problem

Step 2: 10 players play both base ball and foot ball.

Step 3: Here we take 10 players out from the base ball, left out is 15 players play only base ball not a foot ball.

Step 4: We take 10 players out from foot ball, the 15 players left play only foot ball and not base ball.

Step 5: The total number of players participating in the tournament are 10 + 15 + 15 = 40. The 40 players will either play base ball or foot ball among the 60 players present.

60 - 40 = 20.

Answer: Hence the 20 players will not participate in both the matches.

Example 2:

The number of students who like cake and ice cream are shown below:

a) How many number of students like cake?
b) How many number of students like ice cream?
c) How many number of students like only cake?
d) How many number of students like only ice cream?
e) How many number of students like both cake and ice cream?

Solution:

After surveying we get the results, shown below

a) The number of students who like cake are : 34
b) The number of students who like ice cream are : 37
c) The number of students who like only cake are : 18
d) The number of students who like only ice cream are : 21
e) The number of students who like both cake and ice cream are : 16