Challenge Level

This problem is in two parts. The first uses logic in the context of English language whereas the second uses logic in the clearer context of mathematics

Part 1. Which of the following are certainly true, which are certainly false. How many statements form equivalent pairs? Are there any parts of the problem which are debatable or unclear?

If my team wins the world cup tomorrow then I'll be happy
tomorrow.

If I am happy tomorrow then my team will win the world cup
tomorrow.

If I am not happy tomorrow then my team will not win the world
cup tomorrow.

If my team does not win the world cup tomorrow then I will not
be happy tomorrow.

If this is maize then it grew from a seed

If this grew from a seed then it is maize

If this did not grow from a seed then it is not maize

If this is not maize then it did not grow from a seed

If Rover is a dog then Rover is an animal

If Rover is not an animal then Rover is not a dog

If Rover is not a dog then Rover is an animal

If Rover is an animal then Rover is a dog

These ideas will help you to understand part 2.

Part 2. In mathematical logic the implication arrows $\Rightarrow$ and $\Leftrightarrow$ are used to connect expressions as follows:

$p\Rightarrow q$ means 'IF $p$ is true THEN $q$ is true.

$p\Leftrightarrow q$ means both $p\Rightarrow q$ AND $q \Rightarrow p$ simultaneously.

Convince yourself that

$$

\left(p\Rightarrow q\right) \Leftrightarrow \left((NOT q) \Rightarrow (NOT p)\right)

$$

The expression on the right is called the contrapositive of the statement on the left. Since they are linked by $\Leftrightarrow$, proving one side will automatically prove the other.

Consider these statements involving positive integers $n$ and $m$.

1: $n+m$ is odd $\Rightarrow n\neq m$.

2: $n+m$ is even $\Rightarrow$ $n$ and $m$ are either both even or both odd

3: $n^2$ is even $\Rightarrow n$ is even.

4: $n^3$ is odd $\Rightarrow n$ is odd.

5: $n$ mod (4) = 2 or 3 $\Rightarrow$ $n$ is not a perfect square.

Write out the contrapositive versions of these statements and use these to prove the statements. You can assume that an even number can be written as $2N$ and an odd number as $2M+1$.