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Exercises 4.3 Exercises
1.
Use truth tables to determine whether or not the following pairs of statements are logically equivalent.
-
“
\(({\neg} P)\lor Q\)” and “
\(P\Rightarrow Q\)”.
-
“
\(P\Leftrightarrow Q\)” and “
\(({\neg} P)\Leftrightarrow ({\neg} Q)\)”.
-
“
\(P\Rightarrow (Q\lor R)\)” and “
\(P\Rightarrow (({\neg} Q)\Rightarrow R)\)”.
-
“
\((P\lor Q)\Rightarrow R\)” and “
\((P\Rightarrow R)\land(Q\Rightarrow R)\)”.
-
“
\(P\Rightarrow (Q\lor R)\)” and “
\((Q\land R)\Rightarrow P\)”.
2.
-
\(8\) is even and
\(5\) is prime.
-
If
\(n\) is a multiple of
\(4\) and
\(6\text{,}\) then it is a multiple of
\(24\text{.}\)
-
If
\(n\) is not a multiple of
\(10\text{,}\) then it is a multiple of
\(2\) but is not a multiple of
\(5\text{.}\)
-
\(3\leq x\leq 6\text{.}\)
-
A real number
\(x\) is less than
\(-2\) or greater than
\(2\) if its square is greater than
\(4\text{.}\)
-
If a function
\(f\) is differentiable everywhere then whenever
\(x\in\mathbb{R}\) is a local maximum of
\(f\) we have
\(f'(x)=0\text{.}\)
3.
Show that the following pairs of statements are logically equivalent using
Theorem 4.2.3.
-
\(P\iff Q\) and
\((\neg P) \iff (\neg Q)\)
-
\(P\implies(Q\lor R)\) and
\(P\implies((\neg Q) \implies R)\)
-
\((P\lor Q)\implies R\) and
\((P\implies R)\land(Q\implies R)\)
