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Question

# In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If x∈A and A∈B then x∈B (ii) If A⊂B and B∈C then A∈C (iii) If A⊂B and B⊂C then A⊂C (iv) If A⊄B and B⊄C then A⊄C (v) If x∈A and A⊄B then x∈B (vi) If A⊂B and x∉B then x∉A.

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Solution

## (i) The statement is false. Take A = {1}, B = {{1}, 2} Now 1∈A and A∈B but 1∉B. (ii) The statement is false Take A = {1}, B = {1, 2}, C = {{1, 2}, 3} Now A⊂B and B∈C but A∉C. (iii) The statement is true. Let x∈A⇒x∈B (∵ A⊂B) ⇒ x∈C (∵ B⊂C) Now x∈A ⇒ x∈C ∴ A⊂C. (iv) The statement is false. Take A = {1, 2} B = {2, 3 }, C = {1, 2, 5} Now A⊄B and B⊄C but A⊂C. (v) The statement is false. Take A = {1, 2} and B = {2, 3, 4, 5} Now 1∈A and A⊄B but 1∉B (vi) The statement is true. Let x∈A ⇒ x∈B (∵ A⊂B) Now x∉B ⇒ x∉A.

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