The main trouble I am currently having in math is knowing when the use cases are appropriate in a proof. I see many videos where they seem to choose a strategy like proof by contrapositive or proof by contradiction, but never quite understand how they came to the conclusion to use that proof strategy.
Here are some examples I have come across and my own solutions to them using the recommended proofs
the product of two odd integers is odd
I used contradiction to solve it
- suppose the product of two odd integers is even
- (2k+1)(2k+1) = 2k
- 2(2k^2 + 2k) + 1 = 2k
- given k is an integer, (2k^2 + 2k) is an integer. Therefore, an even number cannot be an odd number
if x^2 is even, x is even
Based on the wikipedia article on contrapositive
- if x is odd, then x^2 is odd
- (2k+1)^2 == 2(2k^2 +2k) + 1
- therefore, if x^2 is even, then x is even
However, what I am wondering is, is there a general principle as to when to use specific strategies for proofs? Eg, if you specifically know a theory is false, do you choose a strategy accordingly? What determines which strategy will be most effective, or is it arbitrary?