If U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , A = 1 , 3 , 5 , 6 and B = 2 , 3 , 5 , 7 , then verify that ( A ∩ B ) ' = A ' ∪ B ' .

Given, U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , A = 1 , 3 , 5 , 6 and B = 2 , 3 , 5 , 7 To verify: ( A ∩ B ) ' = A ' ∪ B ' Here, LHS = ( A ∩ B ) ' So, A ∩ B = 1 , 3 , 5 , 6 ∩ 2 , 3 , 5 , 7 = 3 , 5 ∴ ( A ∩ B ) ' = U - A ∩ B = 1 , 2 , 4 , 6 , 7 , 8 , 9 ⇒ LHS = 1 , 2 , 4 , 6 , 7 , 8 , 9 Here, RHS = A ' ∪ B ' Now, A ' = U - A = 2 , 4 , 7 , 8 , 9 and B ' = U - B = 1 , 4 , 6 , 8 , 9 ∴ A ' ∪ B ' = 2 , 4 , 7 , 8 , 9 ∪ 1 , 4 , 6 , 8 , 9 = 1 , 2 , 4 , 6 , 7 , 8 , 9 It shows that, LHS = RHS Therefore, ( A ∩ B ) ' = A ' ∪ B ' .

If n ( A ) = 20 , n ( B ) = 16 and n ( A ∪ B ) = 30 , find n ( A ∩ B ) .

Given, n ( A ) = 20 , n ( B ) = 16 and n ( A ∪ B ) = 30 To find: n ( A ∩ B ) . We know that, n ( A ∪ B ) = n A + n B - n ( A ∩ B ) ⇒ 30 = 20 + 16 - n ( A ∩ B ) ⇒ n ( A ∩ B ) = 36 - 30 ⇒ n ( A ∩ B ) = 6 Therefore, the required answer is 6 .

If n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 , show that n ( A ∪ B ) = 30 .

Given, n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 To show that, n ( A ∪ B ) = 30 . We know that, n ( A ∪ B ) = n ( A - B ) + n ( B - A ) + n ( A ∩ B ) ⇒ n ( A ∪ B ) = 15 + 10 + 5 ⇒ n ( A ∪ B ) = 25 + 5 ⇒ n ( A ∪ B ) = 30 Hence, proved.

If n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 , show that n ( A ) ' = 15 .

Given, n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 To show that: n ( A ) ' = 15 . We know that, n ( B - A ) = n ( A ∪ B ) - n ( A ) ⇒ 10 = 30 - n ( A ) ⇒ n ( A ) = 30 - 10 ⇒ n ( A ) = 20 If universal set U is finite and A is a subset, then n ( A ) + n A ' = n ( U ) ∴ n ( A ) + n A ' = n ( U ) ⇒ 20 + n A ' = 35 ⇒ n A ' = 35 - 20 ⇒ n A ' = 15 Hence, proved.

If n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 , show that n ( B ) = 15 .

Given, n ( U ) = 35 , n ( A - B ) = 15 , n ( B - A ) = 10 and n ( A ∩ B ) = 5 To show that: n ( B ) = 15 . We know that, ∴ n ( B - A ) = n ( B ) - n ( A ∩ B ) ⇒ 10 = n ( B ) - 5 ⇒ n ( B ) = 10 + 5 ⇒ n ( B ) = 15 Hence, proved.

There are 10 elements in set A , 7 elements in set B and 14 elements in set A ∪ B . Are the sets A and B disjoint or overlapping? Also, find n ( A ∩ B ) .

Given, Number of elements in the set A = n ( A ) = 10 Number of elements in the set B = n ( B ) = 7 Number of elements in the set A ∪ B = n ( A ∪ B ) = 14 Two sets are said to be disjoint sets, if they have no element in common. Two sets A and B are said to be overlapping, if they contain at least one element in common. In the given sets the number of elements is less than the total number of distinct elements. Hence, the given sets A and B are overlapping sets. Now, we need to find, n ( A ∩ B ) . We know that, n ( A ∪ B ) = n A + n B - n ( A ∩ B ) ⇒ 14 = 10 + 7 - n ( A ∩ B ) ⇒ n ( A ∩ B ) = 17 - 14 ⇒ n ( A ∩ B ) = 3 Therefore, the required answer is 3 .

E M B I B E

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise

S.N. Chhibber Solutions for Chapter: Operations on Sets, Exercise 3: Exercise

Author:S.N. Chhibber

S.N. Chhibber Mathematics Solutions for Exercise - S.N. Chhibber Solutions for Chapter: Operations on Sets, Exercise 3: Exercise

Attempt the practice questions on Chapter 1: Operations on Sets, Exercise 3: Exercise with hints and solutions to strengthen your understanding. Joy Of Mathematics 8 solutions are prepared by Experienced Embibe Experts.

Questions from S.N. Chhibber Solutions for Chapter: Operations on Sets, Exercise 3: Exercise with Hints & Solutions

MEDIUM

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 8

If $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = \{1, 3, 5, 6\}$ and $B = \{2, 3, 5, 7\}$ , then verify that $(A \cap B)^{'} = A^{'} \cup B^{'}$ .

MEDIUM

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 9

If $U = {1, 2, 3, 4, . . . 15}, A = \{multiples of 2\}$ and $B = \{multiples of 3\}$ , then verify that $(A \cup B)^{'} = A^{'} \cap B^{'}$ .

MEDIUM

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 9

If $U = {1, 2, 3, 4, . . . 15}, A = \{multiples of 2\}$ and $B = \{multiples of 3\}$ , then verify that ${(A \cap B)}^{'} = A^{'} \cup B^{'}$ .

EASY

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 10

If $n (A) = 20, n (B) = 16$ and $n (A \cup B) = 30,$ find $n (A \cap B)$ .

EASY

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 11

If $n (U) = 35, n (A - B) = 15, n (B - A) = 10$ and $n (A \cap B) = 5$ , show that $n (A \cup B) = 30$ .

MEDIUM

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 11

If $n (U) = 35, n (A - B) = 15, n (B - A) = 10$ and $n (A \cap B) = 5$ , show that $n (A)^{'} = 15$ .

EASY

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 11

If $n (U) = 35, n (A - B) = 15, n (B - A) = 10$ and $n (A \cap B) = 5$ , show that $n (B) = 15$ .

MEDIUM

8th CBSE

IMPORTANT

Joy Of Mathematics 8>Chapter 1 - Operations on Sets>Exercise>Q 12

There are $10$ elements in set $A, 7$ elements in set $B$ and $14$ elements in set $A \cup B$ . Are the sets $A$ and $B$ disjoint or overlapping? Also, find $n (A \cap B)$ .

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