ICSE Class 6 Maths Chapter 28 Operations on Sets

Chapter 28: Operations On Sets

28.1 Introduction

Operations on sets means combining two or more sets to get a single set satisfying the given conditions.

Some combining operations discussed in this chapter are:

(i) Union of sets

(ii) Intersection of sets

(iii) Difference of two sets

28.2 Union Of Sets

Union of two given sets is the smallest set that contains all the elements of the two given sets.

To find the union of two given sets, the smallest set with the elements of both the sets is so formed that every element of the two sets is present in this new set but no element is repeated.

For example:

Let set A = {2, 4, 5, 6}

and set B = {4, 6, 7, 8}

Taking every element of the two sets A and B without repeating any of them, we get a new set = {2, 4, 5, 6, 7, 8}.

This new set contains all the elements of set A and all the elements of set B, with no repetition of elements, and is named Union of sets A and B.

The symbol used for the union of two sets is "\(\cup\)".

Therefore, (i) Union of sets A and B = A union B = \(A \cup B\)

(ii) Union of sets P and Q = \(P \cup Q\) and so on.

Example 1

If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8},

find: (i) union of sets P and Q. (ii) \(P \cup R\). (iii) \(Q \cup R\).

Solution

(i) Union of sets P and Q = \(P \cup Q\)

= The smallest set that contains all the elements of set P and all the elements of set Q, without repetition

= {0, 2, 3, 4, 5, 6, 7, 9, 12} (Ans.)

Similarly,

(ii) \(P \cup R\) = Union of sets P and R = {2, 3, 4, 5, 6, 7, 8} (Ans.)

(iii) \(Q \cup R\) = Union of sets Q and R = {0, 2, 3, 4, 6, 8, 9, 12} (Ans.)

Teacher's Note

Union of sets is like combining ingredients from two recipes to make something new without using any ingredient twice. This helps us understand how to organize different groups of items.

28.3 Intersection Of Sets

Intersection of two given sets is the largest set that contains all the elements which are common to both the sets.

For example:

Let set A = {2, 3, 4, 5, 6}

and set B = {3, 5, 7, 9}

The set containing the elements common to the two sets A and B, i.e. {3, 5}, is the Intersection of sets A and B.

The symbol used for the intersection of two sets is "\(\cap\)".

Therefore, (i) Intersection of sets A and B = A intersection B = \(A \cap B\)

(ii) Intersection of sets P and Q = \(P \cap Q\) and so on.

Example 2

If set A = {4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15, 18}

and set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Find: (i) intersection of sets A and B. (ii) \(B \cap C\) (iii) \(A \cap C\).

Solution

(i) Intersection of sets A and B = \(A \cap B\)

= Set of all the elements which are common to set A and set B

= {6, 12} (Ans.)

Similarly,

(ii) \(B \cap C\) = Intersection of sets B and C = {3, 6, 9} (Ans.)

(iii) \(A \cap C\) = Intersection of sets A and C = {4, 6, 8, 10} (Ans.)

If the given sets are in description form or set-builder form, convert them into roster form and then do the required operation.

Example 3

If P = {multiples of 3 between 1 and 20}

and Q = {even natural numbers upto 15}

Find: (i) \(P \cup Q\) (ii) \(P \cap Q\)

Solution

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

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

\(\therefore\) (i) \(P \cup Q\) = Union of sets P and Q

= The smallest set containing all the elements of set P and all the elements of set Q, without repetition of any element.

= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18} (Ans.)

(ii) \(P \cap Q\) = Intersection of sets P and Q

= The largest set containing only those elements that are common to both the given sets.

= {6, 12} (Ans.)

Teacher's Note

Intersection is like finding common interests between two friends, while union is combining all interests. These concepts help us classify and organize information.

Example 4

If M = The empty set and N = {0, \(\triangle\), O}, find:

(i) \(M \cup N\) and (ii) \(M \cap N\)

Solution

Given: M = {} and N = {0, \(\triangle\), O}

\(\therefore\) (i) \(M \cup N\) = {0, \(\triangle\), O} = N (Ans.)

and (ii) \(M \cap N\) = {} = \(\phi\) (Ans.)

\(A \cap \phi = \phi\), i.e. intersection of any set with the empty set, is always the empty set. And, \(A \cup \phi = A\), i.e. union of any set with the empty set, is always the set itself.

Exercise 28(A)

1. State true or false:

(i) If A = {5, 6, 7} and B = {6, 8, 10, 12}, \(A \cup B\) = {5, 6, 7, 8, 10, 12}.

(ii) If P = {a, b, c} and Q = {b, c, d}, P intersection Q = {b, c}.

(iii) Union of two given sets is the set of elements which are common to both the sets.

(iv) Two disjoint sets have at least one element in common.

(v) Two overlapping sets have all elements in common.

(vi) If two given sets have no element common to the two sets, the sets are said to be disjoint.

(vii) If A and B are two disjoint sets, \(A \cap B\) = {}, the empty set.

(viii) If M and N are two overlapping sets, intersection of sets M and N is not the empty set.

2. Let A, B and C be three sets such that:

set A = {2, 4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15}

and set C = {1, 4, 7, 10, 13, 16}.

Find: (i) \(A \cup B\) (ii) \(B \cap C\) (iii) \(B \cap A\) (iv) \(B \cup A\) (v) \(B \cup C\)

Is \(A \cup B = B \cup A\)?

Is \(B \cap C = B \cup C\)?

3. If A = {2, 3, 4, 5}, B = {1, 3, 5, 7} and C = {4, 5, 6, 7}; find:

(i) \(A \cup B\) (ii) \(A \cup C\) (iii) \((A \cup B) \cap (A \cup C)\)

4. If A = {a, b, c, d}, B = {c, d, e, f} and C = {b, d, f, g}; find:

(i) \(A \cap B\) (ii) \(A \cap C\) (iii) \((A \cap B) \cup (A \cap C)\)

5. Let A = Set of natural numbers less than 8

B = {even natural numbers less than 12}

C = {multiples of 3 between 5 and 15}

and D = {multiples of 4 greater than 6 and less than 20}. Find:

(i) \(B \cup C\) (ii) \(A \cup D\) (iii) \(C \cup D\) (iv) \(A \cap C\) (v) \((B \cap C) \cup A\) (vi) \((D \cup A) \cap B\)

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