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 ⊆c B and B ⊆c C, then A ⊆c C (v) If x ∈ A and A ⊆c B, then x ∈ B (vi) If A ⊆ B and x ∉ B, then x ∉ A

(i) False
Let A = {1, 2} and B = {{1}, {1, 2}, 3}
2 ∈ {1, 2} and {1, 2} ∈ {{1}, {1, 2}, 3}
Now, ∴ A ∈ B
However, 2 ∉ {{1}, {1, 2}, 3}
(ii) False. As A ⊆ B, B ∈ C
Let A = {2}, B = {0, 2}, and C = {1, {0, 2}, 3}
However, A ∉ C
(iii) True
Let A ⊆ B and B ⊆ C. Let x ∈ A ⇒ x ∈ B [∵ A ⊆ B] ⇒ x ∈ C [∵ B ⊆ C]
∴ A ⊆ C
(iv) False
As, A ⊈ B and B ⊈ C
Let A = {1, 2}, B = {0, 6, 8}, and C = {0, 1, 2, 6, 9}
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