Question

Let Universal set U = {Natural numbers less than or equal to 30}. Let A and B be two subsets of U defined as A = {Even numbers less than or equal to 30}, and B = {All the factors of 30}. Find n(P(B)), n(AB), n(AB), n(P(AB)), n(P(AB)).​

Answer 1

SOLUTION

GIVEN

Let Universal set U = {Natural numbers less than or equal to 30}.

Let A and B be two subsets of U defined as

A = {Even numbers less than or equal to 30}

B = {All the factors of 30}.

TO DETERMINE

n(P(B)) , n(A' ∩ B'), n(A' ∪ B), n(P(A∩B)), n(P(A∪B)').

EVALUATION

Here it is given that

Universal set U = {Natural numbers less than or equal to 30}.

U = { 1 , 2 , 3 ,..., 30 }

Let A and B be two subsets of U defined as

A = {Even numbers less than or equal to 30}

A = { 2 , 4 , 6 , ... , 30 }

B = {All the factors of 30}.

B = { 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 }

(i) Since B contains 8 elements

So the number of elements in power set of B

$= {2}^{8}$

$= 256$

n(P(B)) = 256

(ii) A'

= { 1 , 3 , 5 ,.. , 29 }

= { Odd numbers less than or equal to 30}

n(B) = 8

n(A' ∩ B')

= n(U) - n(A∪B)

= n(U) - n(A) - n(B) + n(A∩B)

= 30 - 15 - 8 + 4

= 11

(iii) A∩B contains 4 contains

So the number of elements in power set of A∩B

$= {2}^{4}$

$= 16$

n(P(A∩B)) = 16

(iv)

n((A∪B)')

= n(U) - n(A∪B)

= n(U) - n(A) - n(B) + n(A∩B)

= 30 - 15 - 8 + 4

= 11

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