- #1
2^Oscar
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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 [tex]\geq[/tex]) 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 [tex]\geq[/tex] 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
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 [tex]\geq[/tex]) 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 [tex]\geq[/tex] 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