Exercise 1.7

Question

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 $-$.
\begin{align*}
D &= \left\{x \mid x \in A 	ext{ and } (x \in B 	ext{ or } x \in Cight\} \,, \
E &= \left\{x \mid (x \in A 	ext{ and } x \in B) 	ext{ or } x \in Cight\} \,, \
F &= \left\{x \in A 	ext{ and } (x \in B \Rightarrow x \in C)ight\} \,.
\end{align*}

Dan Whitman · Accepted Answer

First, we obviously have
\begin{align*}
D &= A \cap (B \cup C) \
E &= (A \cap B) \cup C \,,
\end{align*}
noting that $D 
eq E$ generally though they appear similar.
Regarding $F$ we have the following sequence of logical equivalences:
\begin{align*}
x \in F &\Leftrightarrow x \in A \land (x \in B \Rightarrow x \in C) \
&\Leftrightarrow x \in A \land (x 
otin B \lor x \in C) \
&\Leftrightarrow x \in A \land \lnot(x \in B \land x 
otin C) \
&\Leftrightarrow x \in A \land \lnot(x \in B - C) \
&\Leftrightarrow x \in A \land x 
otin B - C \
&\Leftrightarrow x \in A - (B - C)
\end{align*}
so that of course $F = A - (B - C)$.

Exercise 1.7

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Exercise 1.7

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