I understand how the steps in a proof are inferred from other steps that are either given or already inferred. Sometimes though, in previous lectures, a professor would begin proving a certain theorem or equation (such as in calculus and statistics) and then at some point that may as well have been arbitrary to me, declared it proven. What decides the last step in a proof and what about it is so special that it “proves” the subject?
In: 4
Many proofs are done by contradiction. Start off assuming the opposite you are trying to prove, then prove that that initial statement is impossible.
eg. an even integer plus an even integer is always an even integer.
1. m,n are even. Start off by contradiction, stating m+n=o where o is odd
2. since m, n are even, they’re divisible by 2. can be written as m=2a, n=2b
3. 2a+2b = o
4. 2(a+b) = o
5. a+b = o/2
well you started off saying m+n is an integer, a and b are integers. So o/2 must be an integer. Therefore o can’t be odd. You’ve proved m+n = even
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