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10_Transcript - 1 Contrapositive 1.1 Welcome Welcome to the...

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1 Contrapositive 1.1 Welcome Welcome to the tenth module in our course on proofs. Before going on, please do the assigned readings. 1.2 How To Use The Contrapositive When trying to prove “ A implies B ” we have two basic techniques so far. The forward-backward method works forward from A and backwards from B until there is a happy meeting in the middle. Proof by contradiction assumes A and NOT B and ﬁnds a contradiction. Contradiction is not the only time we have seen a negated conclusion. Very early on we proved that the contrapositive NOT B NOT A is logically equivalent to A B . This gives us a third approach. Apply the forward-backward method to the statement NOT B NOT A . 1.3 When To Use The Contrapositive This is very similar to contradiction. Use the contrapositive when the statement NOT B or the statement NOT B gives you useful information. This is most likely to occur when A or B contains a negation. When both A and B contain negations, it is highly likely that using the contrapositive will be productive. 1.4 Reading a Proof That Uses the Contrapositive Consider the following proposition. Proposition 1. If x is a real number such that x 3 - 2 x 2 0 then x 2 . Consider the following proof. For reference purposes, each sentence of the proof is written on a separate line. Proof.

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10_Transcript - 1 Contrapositive 1.1 Welcome Welcome to the...

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