Chapter Review

Complete a truth table for the statement $\neg p \to (q \wedge r)$.

Are the statements $p \to (q \vee r)$ and $(p \to q) \vee (p \to r)$ logically equivalent? Explain your answer.

Is the following a valid deduction rule? Explain.

Write the negation, converse and contrapositive for each of the statements below.

1. If the power goes off, then the food will spoil.

2. If the door is closed, then the light is off.

3. $\forall x (x \lt 1 \to x^2 \lt 1)$.

4. For all natural numbers $n$, if $n$ is prime, then $n$ is solitary.

5. For all functions $f$, if $f$ is differentiable, then $f$ is continuous.

6. For all integers $a$ and $b$, if $a\cdot b$ is even, then $a$ and $b$ are even.

7. For every integer $x$ and every integ…

Consider the statement: for all integers $n$, if $n$ is even and $n \le 7$ then $n$ is negative or $n \in \{0,2,4,6\}$.

1. Is the statement true? Explain why.

2. Write the negation of the statement. Is it true? Explain.

3. State the contrapositive of the statement. Is it true? Explain.

4. State the converse of the statement. Is it true? Explain.

Consider the statement: $\forall x (\forall y (x + y = y) \to \forall z (x\cdot z = 0))$.

1. Explain what the statement says in words. Is this statement true? Be sure to state what you are taking the universe of discourse to be.

2. Write the converse of the statement, both in words and in symbols. Is the converse true?

3. Write the contrapositive of the statement, both in words and in symbols. Is the contrapositive true?

4. Write the negation of the statement, both in words…

Write each of the following statements in the form, “if …, then ….” Careful, some of the statements might be false (which is alright for the purposes of this question).

1. To lose weight, you must exercise.

2. To lose weight, all you need to do is exercise.

3. Every American is patriotic.

4. You are patriotic only if you are American.

5. The set of rational numbers is a subset of the real numbers.

6. A number is prime if…

Simplify the following.

1. $\neg (\neg (p \wedge \neg q) \to \neg(\neg r \vee \neg(p \to r)))$.
2. $\neg \exists x \neg \forall y \neg \exists z (z = x + y \to \exists w (x - y = w))$.