Question

Show that AU(B-C)=(AUB)-(C-A)

Answer 1

To prove, A U (B - C) = (A U B) - (C - A)

Proof:

L.H.S. = A U (B - C)

= A U (B ∩ Cᶜ)

= (A U B) ∩ (A U Cᶜ)

= (A U B) ∩ (Aᶜ ∩ C)ᶜ

= (A U B) - (Aᶜ ∩ C)

= (A U B) - (C ∩ Aᶜ)

= (A U B) - (C - A) = R. H. S.

Hence, proved.

Rules:

1. A - B = A ∩ Bᶜ

2. A U (B ∩ C) = (A U B) ∩ (A U C)

3. A ∩ B = B ∩ A

Homeworks for you:

1. A ∩ (B Δ C) = (A ∩ B) Δ (A ∩ C)

2. (A - C) U (B - C) = (A U B) - S

3. A ∩ (B - C) = (A ∩ B) - (A ∩ C)

Answer 2

L.H.S. = A U (B - C)

= A U (B ∩ Cᶜ)

= (A U B) ∩ (A U Cᶜ)

= (A U B) ∩ (Aᶜ ∩ C)ᶜ

= (A U B) - (Aᶜ ∩ C)

= (A U B) - (C ∩ Aᶜ)

= (A U B) - (C - A) = R. H. S.