An introduction to 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…

Suppose that you would like to prove the following implication:

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

Write out the beginning and end of the argument if you were to prove the statement,

1. Directly

2. By contrapositive

3. By contradiction

You do not need to provide details for the proofs (since you do not know what solitary means). However, make sure that you provide the first few and last few lin…

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. Pretend 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, th…

Prove that $\sqrt 3$ is irrational.

Prove the statement: For all integers $n$, if $5n$ is odd, then $n$ is odd. Clearly state the style of proof you are using.

Prove that $\log(7)$ is irrational.