Problem 9
Write the following as English sentences. Say whether they are true or false. $$ \forall n \in \mathbb{Z} . \exists m \in \mathbb{Z} \cdot m=n+5 $$
Problem 9
Negate the following sentences. If \(\sin (x)<0\), then it is not the case that \(0 \leq x \leq \pi\).
Problem 10
Negate the following sentences. If \(f\) is a polynomial and its degree is greater than \(2,\) then \(f^{\prime}\) is not constant.
Problem 10
Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. \((\mathbb{R} \times \mathbb{N}) \cap(\mathbb{N} \times \mathbb{R})=\mathbb{N} \times \mathbb{N}\)
Problem 10
Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " The discriminant is negative only if the quadratic equation has no real solutions.
Problem 10
Decide whether or not the following pairs of statements are logically equivalent. \((P \Rightarrow Q) \vee R\) and \(\sim((P \wedge \sim Q) \wedge \sim R)\)
Problem 10
Translate each of the following sentences into symbolic logic. If \(\sin (x)<0,\) then it is not the case that \(0 \leq x \leq \pi\)
Problem 11
Suppose \(P\) is false and that the statement \((R \Rightarrow S) \Leftrightarrow(P \wedge Q)\) is true. Find the truth values of \(R\) and \(S\). (This can be done without a truth table.)
Problem 11
\((\sim P) \wedge(P \Rightarrow Q)\) and \(\sim(Q \Rightarrow P)\)
Problem 11
Negate the following sentences. You can fool all of the people all of the time.
