ICSE Class 8 Maths Sets Chapter 03 Venn Diagrams

Venn Diagrams

Topics covered:

Venn Diagrams and Set Operations

Properties of Cardinal Numbers

Operations on Three Sets

Associative Laws

Distributive Laws

De Morgan's Laws

Introduction

John Venn (1834-1923), an English mathematician, introduced the concept of pictorially representing sets by elements bounded within a closed figure. This is why these geometrical figures are known as Venn diagrams.

Some of the basic facts about Venn diagrams are:

1. The universal set is represented by a rectangle. All the elements of the sets under consideration lie in the interior of this rectangle.

2. A set is represented by a circle or an ellipse. All the elements of the set lie in the interior of this circle and are written inside the circle. The name and cardinal number of the set are written on the boundary of the circle.

3. If two circles of equal radii describe two sets, this does not necessarily mean that the sets are equal. The boundary of a circle only describes the limits of the set, just as curly brackets do, in set notation. Equal sets are represented by the same circle, as shown below.

4. To highlight the relation between sets, or to describe the result of an operation on sets, particular areas are shaded.

Venn Diagrams and Set Operations

Disjoint Sets

Let \(\xi = \{x \mid x \text{ is a letter of the English alphabet}\}\),

A = {a, e, i, o, u}, and B = {p, q, r, s, t}

Now observe the following operations represented by Venn diagrams:

A ∪ B

{a, e, i, o, u, p, q, r, s, t}

A ∩ B

{} or ∅

A - B

{a, e, i, o, u}

B - A

{p, q, r, s, t}

A'

{b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}

B'

{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, u, v, w, x, y, z}

A' - B

{b, c, d, f, g, h, j, k, l, m, n, v, w, x, y, z}

B' - A

{b, c, d, f, g, h, j, k, l, m, n, v, w, x, y, z}

Intersecting Sets

Let \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}\),

A = {3, 4, 5, 6, 7, 8, 9}, and

B = {7, 8, 9, 10, 11, 12, 13}

Now observe the following operations represented by Venn diagrams:

A ∪ B

{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

A ∩ B

{7, 8, 9}

A - B

{3, 4, 5, 6}

B - A

{10, 11, 12, 13}

A'

{1, 2, 10, 11, 12, 13, 14, 15}

B'

{1, 2, 3, 4, 5, 6, 14, 15}

A' - B

{1, 2, 14, 15}

B' - A

{1, 2, 14, 15}

Subset

Let \(\xi = \{x \mid x \text{ is an even number, } x \leq 20 \text{ and } x \in \mathbb{N}\}\),

A = {x | x is a multiple of 4, x ≤ 20 and x ∈ ℕ}, and

B = {x | x is a multiple of 8, x ≤ 20 and x ∈ ℕ}

Now observe the following operations represented by Venn diagrams:

A ∪ B

{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

A ∩ B

{7, 8, 9}

A - B

{1, 2, 14, 15}

B - A

{1, 2, 14, 15}

A'

{1, 2, 10, 11, 12, 13, 14, 15}

B'

{1, 2, 3, 4, 5, 6, 14, 15}

Teacher's Note

Venn diagrams are like visual maps that help organize information. Think of using circles to sort your music playlist into categories like pop, rock, and indie - some songs fit in multiple categories, just like elements can belong to multiple sets.

Properties of Cardinal Numbers

Let \(\xi = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\),

A = {1, 2, 3, 4, 5, 6, 7}, and

B = {5, 6, 7, 8, 9}

We shall now verify the properties of cardinal numbers by constructing Venn diagrams for the given sets for different operations.

1. n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

The last area n(A ∩ B) has been taken twice in the union of A and B. Thus, n(A ∩ B) is subtracted once.

2. (i) n(A - B) = n(A) - n(A ∩ B)

(ii) n(B - A) = n(B) - n(A ∩ B)

3. (i) n(A') = n(ξ) - n(A)

(ii) n(A ∪ B)' = n(ξ) - n(A ∪ B)

(iii) n(A ∩ B)' = n(ξ) - n(A ∩ B)

Teacher's Note

Cardinal numbers count the total elements in sets, similar to how you count the total students in your class versus just those in the math club - you need to be careful not to count students twice if they're in both groups.

Operations on Three Sets

Let us now consider 3 sets.

Example 1: From the Venn diagram given below, find the results of the following operations and show the result in a new Venn diagram by shading the appropriate area.

(i) A ∪ B ∪ C

A ∪ B ∪ C is represented in the Venn diagram by the total area bound by the sets A, B and C. Thus A ∪ B ∪ C = {1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 17, 20}

(ii) A ∩ B ∩ C

A ∩ B ∩ C is represented in the Venn diagram by the area that is common to all the three sets A, B, and C. Thus A ∩ B ∩ C = {11}

(iii) (A ∪ B) ∩ C

First we find the total area bound by sets A and B. A ∪ B = {3, 5, 7, 8, 9, 10, 11, 12, 13, 15, 20}. Then we find the area that is common to A ∪ B and set C.

Thus (A ∪ B) ∩ C = {8, 11, 12, 13}

(iv) (A ∪ B) - C

We have identified the area that represents (A ∪ B) in the above example. From this area, we exclude the area that is common with set C. Thus (A ∪ B) - C = {3, 5, 7, 9, 10, 15, 20}

Teacher's Note

Working with three sets is like organizing a social event where you track students by grade level, club membership, and sports participation - understanding overlaps between all three categories helps you make better decisions.

Associative Laws

1. Union of sets is associative.

(A ∪ B) ∪ C = A ∪ (B ∪ C)

2. Intersection of sets is associative.

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Teacher's Note

Associative laws show that when combining multiple sets, the grouping doesn't matter - whether you combine the first two sets first or the last two first, you get the same result, just like it doesn't matter if you add numbers (1 + 2) + 3 or 1 + (2 + 3).

Distributive Laws

1. Union of sets is distributive.

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

2. Intersection of sets is distributive.

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Teacher's Note

Distributive laws show how union and intersection interact, similar to how multiplication distributes over addition in arithmetic (a × (b + c) = (a × b) + (a × c)).

De Morgan's Laws

1. The complement of unions is the intersection of the complements.

(A ∪ B)' = A' ∩ B'

2. The complement of intersections is the union of the complements.

(A ∩ B)' = A' ∪ B'

Teacher's Note

De Morgan's Laws help us understand complements in set theory - if you want to find who is NOT in either club A or club B, it's the same as finding who is in neither club A nor club B.

Word Problems

Example 2: In a colony of 178 houses, 108 households own colour televisions and 96 households own music systems. If 28 households own neither a colour TV nor a music system, find how many households own both colour televisions and music systems.

Given n(T) = 108, n(M) = 96, n(ξ) = 178 and n(T ∪ M)' = 28

n(T ∪ M) = n(ξ) - n(T ∪ M)' = 178 - 28 = 150

n(T ∩ M) = n(T) + n(M) - n(T ∪ M) = 108 + 96 - 150 = 54

Thus, 54 households own both colour televisions and music systems.

Example 3: 75% of students in a class chose to participate in a Drama Fest and 50% of the students chose to participate in a Sports Meet. If 24 students have chosen to attend both and 6 students have chosen to attend neither of the two events, how many students study in that class?

Let there be x number of students in the class.

Given: Number of students who participate in both drama as well as sports = n(D ∩ S) = 24

Number of students who participate neither in drama nor in sports = n(D ∪ S)' = 6

Now D = 75% of x = \[\frac{75x}{100} = \frac{3x}{4}\]

Thus, number of students who participate only in drama but not in sports = n(D - S) = \[\frac{3x}{4} - 24\]

Now S = 50% of x = \[\frac{50x}{100} = \frac{x}{2}\]

Thus, number of students who participate only in sports but not in drama = n(S - D) = \[\frac{x}{2} - 24\]

n(ξ) = x = n(D ∩ S) + n(D ∪ S)' + n(D - S) + n(S - D)

\[= 24 + 6 + \frac{3x}{4} - 24 + \frac{x}{2} - 24\]

\[\Rightarrow x = \frac{3x}{4} + \frac{x}{2} + 30 - 48 \Rightarrow x - \frac{3x}{4} - \frac{x}{2} = -18\]

\[\Rightarrow \frac{4x - 3x - 2x}{4} = -18 \Rightarrow \frac{-x}{4} = -18\]

\[\Rightarrow x = 18 \times 4 = 72\]

(multiplying both sides by -4)

Thus, n(ξ) = 72

Thus, there are 72 students in the class.

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