Q. 55.0( 4 Votes )

# If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

Answer :

Given: A × B ⊆ C × D and A × B ≠ ϕ

Need to prove: A ⊆ C and B ⊆ D

Let us consider, (x, y) (A × B) ---- (1)

⇒ (x, y) (C × D) [as A × B ⊆ C × D] ---- (2)

From (1) we can say that,

x A and y B ---- (a)

From (2) we can say that,

x C and y D ---- (b)

Comparing (a) and (b) we can say that,

⇒ x A and x C

⇒ A ⊆ C

Again,

⇒ y B and y D

⇒ B ⊆ D [Proved]

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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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If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

RS Aggarwal - Mathematics(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.

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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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