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in Sets, Relations and Functions by (51.9k points)

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

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Best answer

(i) A ∩ B = A ∩ C need not imply B = C.

Consider, A = {1, 2}

B = {2, 3}

C = {2, 4}

Now,

A ∩ B = {2}

A ∩ C = {2}

Thus, A ∩ B = A ∩ C, here B is not equal to C

(ii) A ⊂ B ⇒ C – B ⊂ C – A

Given as: A ⊂ B

To prove: C–B ⊂ C–A

Consider x ∈ C– B

⇒ x ∈ C and x ∉ B [by definition C–B]

⇒ x ∈ C and x ∉ A

⇒ x ∈ C–A

Hence x ∈ C–B ⇒ x ∈ C–A. This is true for all x ∈ C–B.

∴ A ⊂ B ⇒ C – B ⊂ C – A

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