Exercise 25.

Let \(A\text{,}\) \(B\) and \(C\) be sets.

  1. Suppose that \(A \subseteq B\) and \(B \subseteq C\text{.}\) Does this mean that \(A \subseteq C\text{?}\) Prove your answer. Hint: to prove that \(A \subseteq C\) you must prove the implication, “for all \(x\text{,}\) if \(x \in A\) then \(x \in C\text{.}\)”

  2. Suppose that \(A \in B\) and \(B \in C\text{.}\) Does this mean that \(A \in C\text{?}\) Give an example to prove that this does NOT always happen (and explain why your example works). You should be able to give an example where \(|A| = |B| = |C| = 2\text{.}\)

in-context