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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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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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