For sets A, B and C using properties of sets , prove that
1. A-(B-C)=(A-B)union (A intersection C)
2.A intersection (B-C)=(A intersection B)-(A intersection C)

Question

For sets A, B and C using properties of sets , prove that
1 A-(B-C)=(A-B)union (A intersection C)
2 A intersection (B-C)=(A intersection B)-(A intersection C)

Lovina Kansal · Accepted Answer

Dear student
We have,1) A-B-C=A-B∩C'    ∵B-C=B∩C'=A∩(B∩C')'      [∵X-Y=X∩Y']=A∩(B'∪C)      [∵(B∩C')'=B'∪(C')'=B'∪C=(A∩B')∪(A∩C)=(A-B)∪(A∩C)2) A∩(B-C)= (A∩B)-(A∩C)Let  x be any arbitary element of A∩(B-C)
Then,⇒x∈A∩(B-C)⇒x∈A and x∈(B-C)⇒x∈A and (x∈B and x∉C)⇒(x∈A and x∈B) and (x∈A and x∉C)⇒x∈(A∩B) and x∉(A∩C)⇒x∈(A∩B)-(A∩C)∴A∩(B-C)⊆(A∩B)-(A∩C)Similarly,(A∩B)-(A∩C)⊆ A∩(B-C)Hence,A∩(B-C)= (A∩B)-(A∩C)
Regards

For sets A, B and C using properties of sets , prove that 1. A-(B-C)=(A-B)union (A intersection C) 2.A intersection (B-C)=(A intersection B)-(A intersection C)

For sets A, B and C using properties of sets , prove that
1. A-(B-C)=(A-B)union (A intersection C)
2.A intersection (B-C)=(A intersection B)-(A intersection C)