Q. 34.4( 5 Votes )

# Let A = {x:x ∈ N}, B = {x:x = 2n, n ∈ N), C = {x:x = 2n – 1, n ∈ N} and, D = {x:x is a prime natural number} Find:

i. A ∩ B

ii. A ∩ C

iii. A ∩ D

iv. B ∩ C

v. B ∩ D

vi. C ∩ D

Answer :

A = All natural numbers i.e. {1, 2, 3…..}

B = All even natural numbers i.e. {2, 4, 6, 8…}

C = All odd natural numbers i.e. {1, 3, 5, 7……}

D = All prime natural numbers i.e. {1, 2, 3, 5, 7, 11, …}

i. A ∩ B

A contains all elements of B.

∴ B ⊂ A

∴ A ∩ B = B

ii. A ∩ C

A contains all elements of C.

∴ C ⊂ A

∴ A ∩ C = C

iii. A ∩ D

A contains all elements of D.

∴ D ⊂ A

∴ A ∩ D = D

iv. B ∩ C

B ∩ C = ϕ

There is no natural number which is both even and odd at same time.

v. B ∩ D

B ∩ D = 2

2 is the only natural number which is even and a prime number.

vi. C ∩ D

C ∩ D = {1, 3, 5, 7…}

Every prime number is odd except 2.

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Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}. Then verify each of the following identities:

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × (B – C) = (A × B) – (A × C)

(iii) (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B)

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If A and B be two sets such that n(A) = 3, n(B) = 4 and n(A ∩ B) = 2 then find.

(i) n(A × B)

(ii) n(B × A)

(iii) n(A × B) ∩ (B × A)

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(i) If A ⊆ B, prove that A × C ⊆ B × C for any set C.

(ii) If A ⊆ B and C ⊆ D then prove that A × C ⊆ B × D.

RS Aggarwal - Mathematics

Using properties of sets prove the statements given

For all sets A and B, (A ∪ B) – B = A – B

Mathematics - ExemplarIf A and B are two sets such that n(A) = 23, n(b) = 37 and n(A – B) = 8 then find n(A ∪ B).

Hint n(A) = n(A – B) + n(A ∩ B) n(A ∩ B) = (23 – 8) = 15.

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If A and B are two sets such than n(A) = 54, n(B) = 39 and n(B – A) = 13 then find n(A ∪ B).

Hint n(B) = n(B – A) + n(A ∩ B) ⇒ n(A ∩ B) = (39 – 13) = 26.

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If n(A) = 3 and n(B) = 5, find:

(i) The maximum number of elements in A ∪ B,

(ii) The minimum number of elements in A ∪ B.

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For any sets A, B and C prove that:

A × (B – C) = (A × B) – (A × C)

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