Chapter 1.1, Problem 5E

Let A be a nonempty collection of sets. Determine the truth of each of the following statements and of their converses:

(a) $x\in {\displaystyle {\cup }_{A\in A}A}\Rightarrow x\in A$ for at least one $A\in A$.

(b) $x\in {\displaystyle {\cup }_{A\in A}A}\Rightarrow x\in A$ for every $A\in A$.

(c) $x\in {\displaystyle {\cap }_{A\in A}A}\Rightarrow x\in A$ for at least one $A\in A$.

(d) $x\in {\displaystyle {\cap }_{A\in A}A}\Rightarrow x\in A$ for every $A\in A$.