### Why do this problem?

This problem takes students' logical thinking one step beyond the logical thinking required to follow direct proofs. It will sharpen their understanding of proof and mathematical thinking to a level beyond that normally required in school mathematics, albeit in a simple context.

### Possible approach

The first part of the problem works very well as a group discussion. Initially students might automatically decide that certain of the statements are true or false. But are they absolutely true or absolutely false ? The discussion should lead the group to understand that the statements are mathematically vague, unclear or depend on a personal opinion. Thus, although in normal everday language the statements would typically be considered unambiguous, mathematically they are unacceptable.

Despite their logical vagueness, the statements are useful to understand the concept of the contrapositive: that a statement $A\Rightarrow B$ is equivalent to the statement $NOT(B)\Rightarrow NOT(A)$.

Of course, to understand these statements, students will really need to understand the meaning of the implication arrows $\Rightarrow$ and $\Leftrightarrow$. A good activitiy is to try to get students to explain really clearly these concepts to each other. Holes in understanding will soon become apparent.

Once students grasp thes points, they can move onto the clearer, more formal mathematics in the second part of the question.

A final, powerful part of this activity is that students should try to explain their results to each other in words. This is a really good device for sharpening up mathematical thinking. Can students explain the contrapositive to the class? Do listeners think that their explanation is clear and simple? Can they explain their proofs in the same way?

Don't forget to marvel at the beautiful simplicity of the contrapositive once the results have been proved!

### Key questions

Why might these statements be unclear? How might we make them clear?
Do you understand the meaning of the arrows exactly?
Can you explain your proofs clearly to an audience?

### Possible extension

Can students create other mathematical statements which can be proved by contrapositive?

Can students create other sets of logical statements as in the first part of the question to test out on their peers?

### Possible support

It is best first to tackle IFFY logic before attempting this question.

Students having difficulty with creating the proofs might benefit from being the 'critical audience' to students who can construct the proofs. Can the solvers convince the audience of their results? Once those having difficulty have heard a couple of proofs, they might more clearly see the way to creating their own proofs.