For three sets A, B, and C, show that (i) A ∩ B = A ∩ C need not imply B = C. (ii) A ⊂ B ⇒ C – B ⊂ C – A
For three sets A, B, and C, show that (i) A ∩ B = A ∩ C need not imply B = C. (ii) A ⊂ B ⇒ C – B ⊂ C – A

Answers:

(i) 

Consider,

A = {1, 2}

B = {2, 3}

C = {2, 4}

A ∩ B = {2}

A ∩ C = {2}

Thus, A ∩ B = A ∩ C and B is not equal to C.

(ii) 

A ⊂ B

C–B ⊂ C–A

Consider,

x ∈ C– B

x ∈ C and x ∉ B

x ∈ C and x ∉ A

x ∈ C–A

x ∈ C–B ⇒ x ∈ C–A

∴ A ⊂ B ⇒ C – B ⊂ C – A