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 know that for any value of $x$ with $x> 1$ then we also have $x^2> 1$, 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 $\quad\quad x^2> 1 \Rightarrow x> 1$

However, we know that for any value of $x$ with $x> 1$ then we also have $x^2> 1$, 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.

Some cards can be linked in more than one way, for example the following statements are both true:

- $n>0 \implies n>1$
- $n>0 \implies n>2$

However there is only one way to arrange all 16 cards so that all 8 statements are simultaneously true. Can you show that there is only one way to do this?

What happens to the statement involving $n^3>5n$ if $n$ is not necessarily positive? Are any of the other statements affected if $n$ and $m$ are not necessarily positive?