Q. 83.5( 47 Votes )

# Show that for any sets A and B,

A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)

Answer :

To Prove: A = (A ∩ B) ∪ (A – B)

Proof: Let X ϵ A

Now, we need to show that X ϵ (A ∩ B) ∪ (A – B)

In Case I,

X ϵ (A∩ B)

⇒ X ϵ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B)

In Case II,

X ∉A ∩ B

⇒ X ∉ B or X ∉ A

⇒ X ∉ B (X ∉ A)

⇒ X ∉ A – B ⊂ (A ∪ B) ∪ (A – B)

∴A ⊂ (A ∩ B) ∪ (A – B) (i)

It can be concluded that, A ∩ B ⊂ A and (A – B) ⊂ A

Thus, (A ∩ B) ∪ (A – B) ⊂ A (ii)

Equating (i) and (ii),

A = (A ∩ B) ∪ (A – B)

Now, we need to show, A ∪ (B – A) ⊂ A ∪ B

Let us assume that,

X ϵ A ∪ (B – A)

X ϵ A or X ϵ (B – A)

⇒ X ϵ A or (X ϵ B and X ∉A)

⇒ (X ϵ A or X ϵ B) and (X ϵ A and X ∉A)

⇒ X ϵ (B ∪A)

∴ A ∪ (B – A) ⊂ (A ∪ B) (iii)

Now, to prove: (A ∪ B) ⊂ A ∪(B – A)

Let y ϵ A∪B

yϵ A or y ϵ B

(y ϵ A or y ϵ B) and (X ϵ A and X ∉A)

⇒ y ϵ A or (y ϵ B and y ∉A)

⇒ y ϵ A ∪ (B – A)

Thus, A ∪ B ⊂ A ∪ (B – A) (iv)

∴Using (iii) and (iv), we get:

A ∪ (B – A) = A ∪ B

Rate this question :

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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