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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) False Let A = {1, 2} and B = {1, {1, 2}, {3}} Now, ∴ A ∈ B However, (ii) False Let As A ⊂ B B ∈ C However, (iii) True Let A ⊂ B and B ⊂ C. Let x ∈ A ∴ A ⊂ C (iv) False Let Accordingly,and . However, A ⊂ C (v) False Let A = {3, 5, 7} and B = {3, 4, 6} Now, 5 ∈ A and A ⊄ B However, 5 ∉ B (vi) True Let A ⊂ B and x ∉ B. To show: x ∉ A If possible, suppose x ∈ A. Then, x ∈ B, which is a contradiction as x ∉ B ∴x ∉ A

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