Question

PROVE THAT a∩(b-c)=(a∩b)-(a∩c)

Answer 1

Solutions:  

⇒ Assume 

a= {1,2,3,4,5}  

b= {3,4,5,6,7}

c= {6,7,8,9}   

LHS= a ∩(b-c) b-c= {3,4,5,6,7}-{6,7,8,9} b-c= {3,4,5}  a ∩ (b-c)= {1,2,3,4,5} ∩{3,4,5}

= {3,4,5} (a∩b)- (a∩c)=∩{3,4,5}  (a∩b)={1,2,3,4,5,}∩ {3,4,5,6,7}  = {3,4,5}

a∩c= {1,2,3,4,5}∩{6,7,8,9} = {0} so - 2 (a∩b)-(a∩c) ={3,4,5}-{0} from 1 

and 3 we get a∩(b-c)= (a∩b)-(a∩c) = {3,4,5}

Answer 2

Answer:

• Let x be an arbitrary element in A ∩ (B-C)

↠ x ∈ A and x ∈ (B-C)

↠ x ∈ A, and x ∈ B and x ∉ C

In either case, this implies x ∈ (A and B) but x ∉   (A and C)

↠ x ∈ (A∩B) - (A∩C)

Hence, A ∩(B-C) ⊂ (A∩B) - (A∩C) — (1)

• Again let y be an arbitrary element in (A∩B) - (A∩C)

↠ y   ∈(A∩B) but y  ∉ (A∩C)

↠ y ∈A, and y ∈  B but y ∈ A and y  ∉ C

↠ y ∈ A ∩(B-C)

So, (A∩B) - (A∩C) ⊂ A ∩(B-C)  — (2)

∴ A ∩(B-C) = (A∩B) - (A∩C)