Chapter Summary

Complete a truth table for the statement $\neg P \imp (Q \wedge R)$.

Are the statements $P \imp (Q \vee R)$ and $(P \imp Q) \vee (P \imp R)$ 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. $\forall x (x \lt 1 \imp x^2 \lt 1)$.

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: $\forall x (\forall y (x + y = y) \imp \forall z (x\cdot z = 0))$.

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…

Simplify the following.

1. $\neg (\neg (P \wedge \neg Q) \imp \neg(\neg R \vee \neg(P \imp R)))$.
2. $\neg \exists x \neg \forall y \neg \exists z (z = x + y \imp \exists w (x - y = w))$.

Suppose you break your piggy bank and scoop up a handful of 22 coins (pennies, nickels, dimes and quarters).

1. Prove that you must have at least 6 coins of a single denomination.

2. Suppose you have an odd number of pennies. Prove that you must have an odd number of at least one of the other types of coins.

3. How many coins would you need to scoop up to be sure that you either had 4 coins that were all the…