ICSE Class 7 Maths Chapter 36 Venn Diagram

Chapter 36 Venn-Diagram

36.1 Basic Idea

John Venn, a British Mathematician, developed the idea of using diagrams (closed geometrical figures) to represent sets. These diagrams are called Venn-diagrams.

Usually, a rectangle is used to represent the universal set (E) and circles or oval shaped figures inside the rectangle are used to represent sets under discussion.

In the adjoining figure, the universal set E is represented by a rectangle, whereas two sets A and B are represented by circles.

The shaded portion shows the set A∩B.

The name of the set is written near the boundary of the figure representing the set and its elements inside the figure.

Teacher's Note

Venn diagrams are like sorting diagrams we use in real life, such as organizing students by whether they play sports or music - it helps us see overlaps clearly.

36.2 To Draw A Venn-Diagram

Example 1

Given, universal set (E) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, set A = {2, 4, 6, 8, 10} and set B = {3, 6, 8, 9}. Draw a suitable Venn-diagram to show the relationship between the given sets.

Solution

Steps: 1. Draw a rectangle to represent the universal set. 2. Since the sets are joint, draw two overlapping circles to represent sets A and B. 3. In the common portion of the two circles drawn, write the elements 6 and 8 which are common to sets A and B. 4. Write the remaining elements (2, 4 and 10) of set A in the remaining portion of circle A and the remaining elements (3 and 9) of set B in the remaining portion of circle B. 5. Finally, write the elements (1, 5 and 7) of the universal set, which are neither in A nor in B, outside the two circles as shown.

Note: To draw a Venn-diagram for a set with a very large number of elements in it, draw a closed figure and write a brief description of the set. Thus, for the set A = {x: x is a student of Sophia School} the Venn-diagram will be as shown alongside.

Teacher's Note

Drawing Venn diagrams teaches us to systematically organize information, just like sorting laundry into piles for different family members.

36.3 Using Venn-Diagrams To Show The Relationship Between The Different Sets

1. The adjoining figure shows two disjoint sets, i.e., the sets which do not have any element in common. Clearly, A∩B = 0.

2. The adjoining figure shows two joint or overlapping sets.

It is clear that: (i) A∩B = portion common to A and B both = portion marked by slant lines. (ii) A - B = portion of A, which is not in B = portion marked by horizontal lines. (iii) B-A = portion of B, which is not in A = portion marked by vertical lines.

3. In the adjoining figure, the names of the two sets A and B are written near the boundary of the same circle, this shows: Set A = Set B.

4. In the adjoining figure, the circle representing set B is completely inside the circle representing set A. This shows that set B is a proper subset of set A, i.e., B⊂A.

5. In the similar manner, study the shaded portion of each of the following Venn-diagrams:

(i) Complement of set A, i.e., A'

(ii) Complement of set B, i.e., B'

(iii) Complement of A union B, i.e., (A∪B)'

(iv) Complement of A intersection B, i.e., (A∩B)'

6. The shaded portion in the given figure shows: A-B or, A∩B' or, only A.

7. The shaded portion in the adjoining figure shows: B-A or, A'∩B or, only B.

Teacher's Note

Understanding these relationships helps us analyze real situations, like how students might overlap in different clubs or activities at school.

36.4 Finding The Elements Of Various Sets From The Given Venn Diagram

Example 2

From the adjoining venn-diagram, find: (i) A∪B (ii) A∩B (iii) A-B (iv) B-A (v) (A∩B)' (vi) (A∪B)'

Solution

(i) A∪B = {elements, which are in A or in B or in both} = {2, 4, 6, 7, 9, 10}

(ii) A∩B = {elements, which are common to both the sets A and B} = {4, 9}

(iii) A - B = {elements, which are in A and not in B} = {2, 6}

(iv) B-A = {elements, which are in B and not in A} = {7, 10}

(v) (A∩B)' = {elements of the universal set, which are not in A∩B} = {1, 2, 3, 5, 6, 7, 8, 10}

(vi) (A∪B)' = {elements of the universal set, which are not in A∪B} = {1, 3, 5, 8}

Example 3

Draw a Venn-diagram to show the relationship between the following sets: Universal set = {integers}, A = {whole numbers} and B = {natural numbers}.

Solution

Since, every natural number is always a whole number, B is a subset of A and so the diagram will be as shown alongside.

Teacher's Note

This example demonstrates how mathematical sets relate to each other, similar to how all students in a chess club are part of the larger group of students in school.

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