Why do 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
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
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
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
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!
Why might these statements be unclear? How might we make them
Do you understand the meaning of the arrows exactly?
Can you explain your proofs clearly to an audience?
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?
It is best first to tackle IFFY
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