Question

prove that : A-(BnC)=(A-B)U(A-C)

Answer 1

U={1,2,3,4,5,6,7}
A={2,3,6}
B={1,4,5,7}
C={1,4,6,3}
BnC={1,4}
A-BnC={2,3,6}
A-B={2,3,6}
A-C={2}
(A-B )U (A-C)={2,3,6}
Hence probed

Answer 2

Answer:  The proof is done below.

Step-by-step explanation:  For any three sets A, B and C, we are given to prove the following :

A - (B ∩ C) = (A - B) U (A - C).

We know that

any two sets P and Q are equal if and only if both are subsets of each other, that is

P ⊂ Q and Q ⊂ P.

Let us consider that

   x ∈ A - (B ∩ C)

⇒x∈A, x ∉ (B ∩ C)

⇒x∈A, (x∉B or x∉C)

⇒(x∈A, x∉B) or (x∈A, x∉C)

⇒x∈(A-B) or x∈(A-C)

⇒x∈(A-B) ∪ (A-C)

So, A - (B n C) ⊂ (A-B) U (A-C).

Again, let

  x∈(A-B) ∪ (A-C)

⇒x∈(A-B) or x∈(A-C)

⇒(x∈A, x∉B) or (x∈A, x∉C)

⇒x∈A, (x∉B or x∉C)

⇒x∈A, x ∉ (B ∩ C)

⇒x ∈ A - (B ∩ C).

So, A - (B ∩ C) ⊂ (A-B) ∪ (A-C).

Therefore, we get

A - (B n C) = (A - B) U (A - C).

Hence proved.