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Free_Kalibri [48]
2 years ago
9

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

Mathematics
1 answer:
patriot [66]2 years ago
5 0

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}\!.

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