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Iffy Logic

Age 14 to 18
Challenge Level
This problem explores the way that mathematical statements link together logically.

For example, for any number $x$ the expressions $x> 1$ and $x^2> 1$ are both mathematical statements which might be true or might be false. However, we always know that if $x> 1$ then $x^2> 1$, whereas it is not always the case that if $x^2> 1$ then $x>1$ (consider $x=-2$, for example). Thus:

It is correct to write $\quad\quad x> 1 \Rightarrow x^2>1$

It is incorrect to write $\quad\quad x^2> 1 \Rightarrow x> 1$

In the interactivity below, there are sixteen statements. Assuming that n and m are positive integers, can you sort them into eight pairs of statements?

In four of the pairs, the implication only works in one direction, whereas in the other four pairs, each statement implies the other.

If you want to work on this away from the computer, you can print out the statements

Is there more than one possible solution? How do you know?

What if $n$ and $m$ were not necessarily positive or not necessarily integers?