Given sets \(A, B\), and \(C\), express each of the following sets in terms of \(A, B\), and \(C\), using the symbols \(\cup, \cap,\) and \(-\) .
\[ D = \{x \mid x \in A \text { and } (x \in B \text { or } x \in C) \}, \]
\[ E = \{x \mid (x \in A \text { and } x \in B) \text { or } x \in C \}, \]
\[ F = \{x \mid x \in A \text { and } (x \in B \Rightarrow x \in C) \}. \]
Solution
Directly translating the logical connectives gives \[ D=A\cap(B\cup C) \] and \[ E=(A\cap B)\cup C. \] For \(F\), use \[ (x\in B\Rightarrow x\in C) \iff (x\notin B\text{ or }x\in C). \] Thus, among points of \(A\), the only points excluded are those lying in \(B-C\). Therefore \[ \boxed{F=A-(B-C).} \] Equivalently, \[ F=(A-B)\cup(A\cap C). \]