Proofs

Consider the statement “for all integers $a$ and $b$, if $a + b$ is even, then $a$ and $b$ are even”

1. Write the contrapositive of the statement.

2. Write the converse of the statement.

3. Write the negation of the statement.

4. Is the original statement true or false? Prove your answer.

5. Is the contrapositive of the original statement true or false? Prove your answer.

6. Is the converse of the original statement true or false? Pro…

For each of the statements below, say what method of proof you should use to prove them. Then say how the proof starts and how it ends. Bonus points for filling in the middle.

1. There are no integers $x$ and $y$ such that $x$ is a prime greater than 5 and $x = 6y + 3$.

2. For all integers $n$, if $n$ is a multiple of 3, then $n$ can be written as the sum of consecutive integers.

3. For all integers $a$ and $b$, if $a^2 + b^2$ is odd, then $a$ or

The game TENZI comes with 40 six-sided dice (each numbered 1 to 6). Suppose you roll all 40 dice.

1. Prove that there will be at least seven dice that land on the same number.

2. How many dice would you have to roll before you were guaranteed that some four of them would all match or all be different? Prove your answer.