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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 \((P\implies R)\land(Q\implies R)\)
