Question

[Q 9] Let A, B and C be sets. Then without putting/ considering any numeral values for A, B, C
show that A U (B n C) = (A U B) N (AUC).
(Understanding)​

Answer 1

Step-by-step explanation:

To prove

A∪(B∩C) = (A∩B)∪(A∩C)

Concept

If X and Y are two sets, then

X⊂Y and Y⊂X ⇔ X = Y

Hence, to prove the given equality it is sufficient to prove that LH S and RHS are subset of each other.

Proof

Let x ∈ A∪(B∩C)

⇒ x ∈ A or x ∈ (B∩C)

⇒ x ∈ A or ( x ∈ B and x ∈ C )

⇒ ( x ∈ A or x ∈ B ) and ( x ∈ A or x ∈ C )

⇒ x ∈ (A∪B) and x ∈ (A∪C)

⇒ x ∈ (A∪B) ∩ (A∪C)

i.e if x ∈ A∪(B∩C), then x ∈ (A∪B) ∩ (A∪C)

So A∪(B∩C) ⊂ (A∪B) ∩ (A∪C) ..........(1)

Let y ∈ (A∪B) ∩ (A∪C)

⇒ y ∈ (A∪B) and x ∈ (A∪C)

⇒ ( y ∈ A or y ∈ B ) and ( y ∈ A or y ∈ C )

⇒ y ∈ A or ( y ∈ B and y ∈ C )

⇒ y ∈ A or y ∈ (B∩C)

⇒ y ∈ A∪(B∩C)

i.e. if y ∈ (A∪B) ∩ (A∪C), then y ∈ A∪(B∩C)

So (A∪B) ∩ (A∪C) ⊂ A∪(B∩C)........(2)

From (1) and (2),

A∪(B∩C) = (A∪B) ∩ (A∪C)

Hence Proved.