ProofTips

# ProofTips - Finding the Next Step Modern Algebra Fall 2011...

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Unformatted text preview: Finding the Next Step Modern Algebra, Fall 2011 It is bound to happen in this class. You understood everything in class and in the text- book, you can follow along when the professor shows a proof on the board, but you’re staring at a homework exercise that says “Prove the following...,” and you have absolutely no idea what to do. Everyone feels this way sooner or later, and even professional mathematicians feel exactly this way about the (much more difficult) problems they work on. So here are some tips, some suggestions for what to do when you get stuck trying to find a proof. 1. Differentiate the hypotheses (which you get to assume) from the conclusion (which you do not get to use, because it is your goal). Key words for hypotheses are “suppose,” “if,” and “let”; key words for the conclusion are “prove” and “show.” 2. Very often the goal structures the proof . That is, the conclusion you are trying to prove will suggest an overall structure or “shell” for your proof. For example, if you want to prove (a) that A ⊆ B or (b) that the function ϕ is 1-1, you know that your proof will probably run like this: Claim: A ⊆ B Claim: ϕ is 1-1....
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ProofTips - Finding the Next Step Modern Algebra Fall 2011...

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