Explanation:
Direct:
Show, without any trick of proving, that the premises lead to the conclusion without any deroute into proof method land.
Contradiction: suppose that the conclusion isn't true. Then show what errors that would bring to the mathematical system. Hence conclude that the conclusion must be true
Contrapositive: to show that the premises lead to the conclusion, show that if the conclusion is false then the premise must be false. Then you can use proof by contradiction on that to show that the conclusion can't be false, it must be true.
Proof by cases: let's say you know you can break up the premise into cases. Then show that each case leads to the conclusion individually, and therefor the premise leads to the conclusion.
Proof for uniqueness: proof that if there are two solutions, then the solutions must be equal. Then you've shown there is only one unique solution
Comments (10)
I find induction proofs to be most fun to prove, but of the ones listed proof by contradiction is always fun to do.
Oh yes! Forgot about that one
We actually have to do a whole unit on proof of induction I’m vce aha it was fun and annoying at the same time
Proof by contradiction proves that root 2 is irrational and quite frankly is like being kicked in the guys. Since you prove someone wrong by doing their methods, it’s the greatest insult. I guess that’s why the guy who proved root 2 is irrational was sent to exile.
However, a close second is proof by induction, as I’ve always liked the method used.
Out of the options, contradiction is my favourite (although, contrapositive proofs have a place near and dear to my heart). On the other hand, my favourite method for proving things is induction. It is like you are playing dominoes, but with statements about integers instead of domino pieces.
Isn't Proof of Contradiction is to show the assumption of the proposition to be false leads to a contradiction?
Nope. Consider some statement of the form:
If P, then Q.
When we do a proof by contradiction, we assume the negation of the statement. If-then statements are negated as follows:
P and not Q.
As you can see, the conclusion Q is negated, while P is practically untouched.
Reply to: League
It's like
P→ Q
We need to show that the negation of (~P v Q) is false. Hence we conclude the opposite must be true.
Reply to: Never
~P v Q is logically equivalent to P —> Q so you are correct. The negation would be P ^ ~Q.
Also notice that an implication statement (P—>Q) is true whenever P (the hypothesis) is false. So what we really do is assume the hypothesis to be true and show Q is false, the only time an implication is false.