Question

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

a) If Fido is our fish's name then Rex is our dog's name.
b) It is false that Fido
is not our fish's name and Rex is not our dog's name.
c) Fido is our fish's name or Rex is our dog's name.

Answer 1

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

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}\!`.