Question

Determine which, if any, of the three statements are equivalent.

Answer 1

Answer:

Statement (b) is equivalent to statement (c), but neither is equivalent to statement (a).

Step-by-step explanation:

Let $F$ denote that "Fido is our fish's name" and let $R$ denote that "Rex is our dog's name". The statements listed in this question would then become:

• (a) $F \implies R$;
  $F$ implies $R$.
• (b) $\lnot ((\lnot F) \land (\lnot R))$;
  not ((not $F$) or (not $R$)).
• (c) $F \lor R$;
  $F$ or $R$.

Apply de Morgan's Law $\lnot (P \land Q) \iff (\lnot P) \lor (\lnot Q)$ to rewrite statement (b). Let $P = (\lnot F)$ and $Q = (\lnot R)$. By de Morgan's Law:

$\begin{aligned} & \lnot ((\lnot F) \land (\lnot R)) \\ \iff \; & \lnot (P \land Q)\\ \iff \; & (\lnot P) \lor (\lnot Q) && (\text{by de Morgan's Law}) \\ \iff \; & (\lnot (\lnot F)) \lor (\lnot (\lnot R)) \\ \iff \; & F \lor R\end{aligned}$.

In other words, statement (b) $\lnot ((\lnot F) \land (\lnot R))$ is equivalent to statement (c) $F \lor R$.

Statement (a) $F \implies R$ isn't equivalent to $F \lor R$.

For example, when $F = \texttt{True}$ and $R = \texttt{False}$, $F \implies R$ would be false since $\texttt{True}$ does not imply $\texttt{False}$. However, for the same $F$ and $R$, $F \lor R$ would be true since $F = \texttt{True}\!$.