Have I Understood the Process of Proof by Contrapositive Correctly?

[2^Oscar]

Hi guys,

I've just started university this week and I've been given a mountain of assignments. One of them has a proof question in it. Since this is an assignment I want to make clear that I don't want help with the actual proof.

In the first part of the question I'm asked to, given a particular inequality (lets call it A), show that another inequality (this one B) is true. This was a trivial proof by induction.

Next I am asked to prove that given inequality B, that inequality A is true. In the workbook it mentions in passing something called the contrapositive which is something I haven't encountered before. I can't get the answer to drop out using the standard induction method so I assume I need to use this new one.

My understanding of the contrapositive from looking at articles on the internet is that, to show A is true given B, I need to contradict the inequality of A (for example if A has < I need to flip it to \geq) and then prove, using basically the same induction process as the first part of the question, that the contradiction of B is true (e.g. if B had sign < I need to show through induction that the same inequality just with a \geq sign is true). I then would state that it is true for the contrapositive hence the statement is true.

I guess my question is; have I correctly understood the process of proof by contrapositive outlined above?

Thanks for the help in advance,
Oscar

 

[HallsofIvy]

Yes, the "contrapositive" of the statement "If B then A" is "if not A then not B" so you would prove "If A then B", using the contrapositive" by starting "if A is not true, then ..." and using that to prove that B is not true.