91Ó°ÊÓ

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. If \(x\) and \(y\) are real numbers and \(5 x=5 y\), then \(x=y\).

\(\sim(P \vee Q)=(\sim P) \wedge(\sim Q)\)

If \(a \in \mathbb{Q}\) then \(5 a \in \mathbb{Q},\) and if \(5 a \in \mathbb{Q}\) then \(a \in \mathbb{Q}\)

Negate the following sentences. For every positive number \(\varepsilon\), there is a positive number \(\delta\) such that \(|x-a|<\delta\) implies \(|f(x)-f(a)|<\varepsilon\).

Write the following as English sentences. Say whether they are true or false. $$ \forall X \in \mathscr{P}(\mathbb{N}), X \subseteq \mathbb{R} $$

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " A function is rational if it is a polynomial.

Translate each of the following sentences into symbolic logic. For every prime number \(p\) there is another prime number \(q\) with \(q>p\).

Use truth tables to show that the following statements are logically equivalent. \(\sim(P \vee Q \vee R)=(\sim P) \wedge(\sim Q) \wedge(\sim R)\)

Negate the following sentences. For every positive number \(\varepsilon\), there is a positive number \(M\) for which \(|f(x)-b|<\varepsilon\) whenever \(x>M\).

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " An integer is divisible by 8 only if it is divisible by 4