Question 438866: I was wondering if someone could give me a proof that I could use for an example? I have to come up with a few of my own and prove (or disprove them) and I am confused on how to exactly prove a proof that I came up with. I am fine following Euclid, I just am confused when I have to prove something on my own. If anyone could give me an example one so I can see how it is done that would be amazing.
Thanks! :)
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! There are very many different types of proofs used in mathematics. When you prove something, you want to make sure that the statement holds for all cases, using whatever techniques there are. Common techniques used to prove something are:
*direct proof (go right into the statement by using pre-established axioms and theorems)
*mathematical induction (if P(1) is true and P(n) implies P(n+1) then P is true for all n)
*contradiction, indirect proof (suppose the opposite is true, find a contradiction)
*proof by exhaustion (self-explanatory; was used to prove the four-color theorem)
*transposition (if you want to prove "P implies Q," you can prove the contrapositive "if not Q, then not P"
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Here are various examples of questions which will require proofs:
1 (introductory) Show that the area of a trapezoid is  .
2 (moderate) A rectangular prism has a fixed surface area, but its dimensions may vary. Prove that the largest possible volume occurs when the rectangular prism is a cube (you don't need multi-variable calculus to optimize the volume, a nice solution comes from using inequalities).
3 (Riemann hypothesis, unsolved) Prove that all non-trivial zeros of the Riemann zeta function have a real part of 1/2. However this problem will be beyond the scope of secondary school classes and most university classes so we won't get to this here.
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