ICSE Class 6 Maths Chapter 30 Venn Diagrams

Chapter 30: Venn Diagrams

Using Geometrical Figures to Represent Sets

Venn-Diagrams

Venn diagrams are closed figures inside which are marked some points. The closed figure represents a set and the points marked inside it represent the elements of the set.

The closed figure given alongside represents the set = { 4, 5, 6, 7, 8, 9 }.

The closed geometrical figures used to represent sets are called Venn diagrams; after the name of an English Mathematician John Venn, who introduced these figures for set representation.

Though there is no restriction on what shape of the figure should be, a circle or a rectangle is most commonly used to represent a set.

Example 1

Represent the following sets by Venn diagrams:

(i) Set A = {p, q, r} (ii) Set B = {a, b, c, d} (iii) Set C = {1, 2, 3, 4}

Solution

(i) A circle labeled A contains points p, q, and r.

(ii) A circle labeled B contains points a, b, c, and d.

(iii) A rectangle labeled C contains the numbers 1, 2, 3, and 4 arranged in a 2x2 grid.

As is clear from Example 1 given above:

1. The name of the set is written near the boundary of the drawn figure.

2. The names of the elements of the set are written near the points that are marked inside the figure.

Sometimes these points are not marked, and only the elements are written inside the closed figure.

Note: If the number of elements is large and it is not possible to enter the names of all the elements inside the figure, the well-defined description of the set is written inside the drawn figure.

For example: If set B = {Indians}, Venn diagram for set B will be as shown alongside:

A circle labeled B contains the text "Indians".

Teacher's Note

Venn diagrams help us visualize which students in a class play basketball and which play football, making it easy to see who plays both sports.

Venn-Diagram to Show the Relationship Between the Given Sets

1. When the sets are disjoint

If the two given sets are disjoint, they are shown by two separate figures drawn side by side.

Example 2

Use a Venn-diagram to show the relationship between sets A and B when:

A = { Natural numbers less than 5 }

and B = { Natural numbers more than 6 }

Solution

First of all, write the given sets in roster form.

Here, A = { 1, 2, 3, 4 }

and B = { 7, 8, 9, ... }

As the sets A and B are disjoint, they are represented by a Venn-diagram as shown alongside:

Two separate circles are drawn. The left circle is labeled A and contains 1, 2, 3, 4. The right circle is labeled B and contains 7, 8, 9, and dots.

Sets A and B have no element in common, so they are disjoint sets.

For any two disjoint sets A and B, A ∩ B = φ, the empty set. Conversely, if A ∩ B = φ, the sets A and B are disjoint.

Teacher's Note

Two friend groups with no members in common are like disjoint sets - they don't overlap at all.

2. When the sets are overlapping

If two given sets are overlapping, they are shown by two intersecting figures. The elements common to both the sets are written inside the part that is common to the figures drawn.

Example 3

Given P = { multiples of 3 less than 20 }

and Q = { multiples of 2 up to 20 }

Show the relationship between these sets by a Venn-diagram.

Solution

On writing the given sets in roster form, we get:

P = { 3, 6, 9, 12, 15, 18 }

and Q = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 }

Clearly, P ∩ Q = { 6, 12, 18 }

Therefore, the two sets are represented by a Venn-diagram as drawn alongside:

Sets P and Q have elements 6, 12 and 18 in common; thus P and Q are overlapping sets.

Two overlapping circles are drawn. The left circle is labeled P and contains 3, 9, 15 on the left side. The overlapping region in the middle contains 6, 12, 18. The right circle is labeled Q and contains 2, 4, 8, 10, 14, 16, 20.

Teacher's Note

A class where some students like both pizza and burgers represents an overlapping set - they share common members.

When all the elements of one set are present in the second set, the Venn-diagram for the two sets is obtained by drawing one circle inside another circle.

For example: Let A = { even numbers up to 10 } and B = { multiples of 4 up to 10 }

Clearly, A = { 2, 4, 6, 8, 10 } and B = { 4, 8 }

Therefore, the Venn-diagram showing the relationship between sets A and B will be as drawn alongside:

A circle labeled A contains the numbers 2, 6, and 10 on the outside. Inside circle A is a smaller circle labeled B containing 4 and 8.

Example 4

Draw a Venn-diagram showing the relationship between sets B and P, where

B = { Birds } and P = { Parrots }

Solution

Draw one circle to represent the set of birds, i.e. set B.

Since all parrots are birds, draw a smaller circle showing set P inside the circle drawn for set B.

A circle labeled B contains a smaller shaded circle labeled P inside it.

Note: In the Venn-diagram drawn above for example 4, the shaded part represents the set of parrots and the unshaded part represents all other birds that are not parrots.

Example 5

Given: G = { girls of class X } and T = { girls who play tennis }.

Draw a Venn-diagram to show the relationship between the given sets G and T.

Solution

It is obvious that all the girls of Class X cannot play tennis and all the girls who play tennis cannot be in Class X.

Therefore, the required Venn-diagram representing the relationship between sets G and T is as shown alongside:

Two overlapping circles are drawn. The left circle is labeled G and the right circle is labeled T. The overlapping region in the middle is shaded with horizontal lines.

In Example 5 given above, the shaded portion, which is common to the sets G and T, represents the girls of Class X who play tennis as well. The unshaded portion in set G represents the girls of Class X who do not play tennis, and the unshaded portion in set T represents the girls who play tennis but are not in Class X.

Teacher's Note

When drawing a Venn diagram for swimmers in your school and tall people, the overlap shows students who are both swimmers and tall.

Summary

1. Drawing Venn diagram when sets A and B are disjoint, i.e. they do not have any element in common:

(i) Two separate circles labeled A and B with no overlap. A ∩ B = The empty set

(ii) Two separate shaded circles labeled A and B. A ∪ B = The shaded region.

2. When sets A and B are overlapping, i.e. the two sets have at least one element in common:

(i) Two overlapping circles labeled A and B. The overlapping region is shaded. A ∩ B = The shaded region

(ii) Two overlapping circles labeled A and B. The entire area of both circles is shaded. A ∪ B = The shaded region.

3. When set A is contained in set B, i.e. all the elements of set A are in set B:

(i) A small shaded circle labeled A inside a larger circle labeled B. A ∩ B

(ii) A large shaded circle labeled B containing circle A inside it. A ∪ B

4. When all the elements of set B are in set A:

(i) A small shaded circle labeled B inside a larger circle labeled A. A ∩ B

(ii) A large shaded circle labeled A containing circle B inside it. A ∪ B

Using A Given Venn Diagram

A given Venn-diagram can be used to find different sets.

Example 6

From the adjacent Venn-diagram, find the following sets:

(i) C (ii) D (iii) C ∩ D (iv) C ∪ D

The diagram shows two overlapping circles. Circle C on the left contains: r, s, p, t, q, k, u, v, m. Circle D on the right contains: n, o, i, j, g, m, k, q. The overlapping region contains: q, k, m.

Solution

(i) C = Set of all elements inside the circle representing set C = { p, q, r, s, t, u, v, m }

(ii) D = Set of all elements inside the circle representing set D = { g, j, k, l, n, o, p, q, m }

(iii) C ∩ D = Set of elements common to both the sets C and D = { p, q, m }

(iv) C ∪ D = Set of elements that are in set C or in set D or in both = { p, q, r, s, t, u, v, m, g, j, k, l, n, o }

Teacher's Note

Reading Venn diagrams is like finding which books are only in your library, which are only in your friend's library, and which books you both have.

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