Using properties of sets prove the statements given:-

(i) For all sets A and B, AU(B-A) = AUB

(ii) For all sets A and B, A-(A-B) = AintersectionB

(iii) For all sets A and B, (AUB)-B = A-B

Question

Using properties of sets prove the statements given:-
(i) For all sets A and B, AU(B-A) = AUB
(ii) For all sets A and B, A-(A-B) = AintersectionB
(iii) For all sets A and B, (AUB)-B = A-B

Anuradha Sharma · Accepted Answer

1)

To show that A u (B - A) = A U B we need to show two
things:

1) A U (B - A) ⊆ A U B

2) A U B ⊆ A U (B - A)

First, let x ∈A U (B - A)
x∈A or x∈B-Aor x∈A∪B or x∈B and x∉Aor
x∈A∪B or x∈BThus x∈A∪BSo
A∪B-A⊆A∪B 1Now let x∈A∪B
⇒x∈A or x∈B and x∉A⇒x∈A or
x∈B-A⇒x∈A∪B-ASo A∪B⊆A∪B-A 2Now
using equation 1 and 2, A∪B=A∪B-AHence proved

In the similar way should try for the other two for your own
practice

Using properties of sets prove the statements given:- (i) For all sets A and B, AU(B-A) = AUB (ii) For all sets A and B, A-(A-B) = AintersectionB (iii) For all sets A and B, (AUB)-B = A-B

Using properties of sets prove the statements given:-

(i) For all sets A and B, AU(B-A) = AUB

(ii) For all sets A and B, A-(A-B) = AintersectionB

(iii) For all sets A and B, (AUB)-B = A-B