Primary reference: Chapter 5 (Contrapositive Proof) and Chapter 6 (Proof by Contradiction) of Book of Proof by Richard Hammack
Content covered in this unit includes:
The contrapositive and converse of a mathematical statement
The recipe for writing a proof by contrapositive:
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Proposition. If $P$, then $Q$.
Proof. We will prove the contrapositive. Suppose $\mathord{\sim} Q$.
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Therefore $\mathord{\sim} P$.
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The recipe for writing a proof by contradiction:
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Proposition. $P$.
Proof. Suppose for the sake of contradiction that $\mathord{\sim} P$.
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Therefore $C$ and $\mathord{\sim} C$, which is a contradiction. We conclude that $P$.
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The recipe for proving a conditional statement by contradiction
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Proposition. If $P$, then $Q$.
Proof. Suppose for the sake of contradiction that $P$ and $\mathord{\sim} Q$.
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Therefore $C$ and $\mathord{\sim} C$, which is a contradiction. We conclude that $P \implies Q$.
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The precise definitions of the terms congruent modulo $n$, rational, irrational