Q. 44.3( 6 Votes )

# (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.

Answer :

(i) Given: A ⊆ B

Need to prove: A × C ⊆ B × C

Let us consider, (x, y) (A × C)

That means, x A and y C

Here given, A ⊆ B

That means, x will surely be in the set B as A is the subset of B and x A.

So, we can write x B

Therefore, x B and y C ⇒ (x, y) (B × C)

Hence, we can surely conclude that,

A × C ⊆ B × C [Proved]

(ii) Given: A ⊆ B and C ⊆ D

Need to prove: A × C ⊆ B × D

Let us consider, (x, y) (A × C)

That means, x A and y C

Here given, A ⊆ B and C ⊆ D

So, we can say, x B and y D

(x, y) (B × D)

Therefore, we can say that, A × C ⊆ 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)

RS Aggarwal - Mathematics

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)

RS Aggarwal - Mathematics

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.

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.

RS Aggarwal - Mathematics