Difference between revisions of "Proofs:Contraposition"
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| − | Let <math> P </math> and <math> Q </math> be propositions such that <math> P \implies Q </math>. Then, the contrapositive of the conditional statement "<math> P \implies Q </math>" (read as "if P, then Q" or "P implies Q") is "<math> \neg Q \implies \neg P </math> (read as "if not Q, then not P" or "not Q implies not P"). The contrapositive is logically equivalent to the original conditional; that is, a conditional and its contrapositive always have the same truth values. For example, the contrapositive of "if <math> x > 0 </math>, then <math> x </math> is positive" is "if <math> x </math> is NOT positive, then <math> x \leq 0 </math>". These two statements are logically equivalent, and are both true. Sometimes, proving the contrapositive of a conditional is easier than proving the conditional itself. For example: Let x be an integer. If <math> x^2 </math> is even, then x is even. While we can attempt to prove this conditional statement directly, it is easier to show that if x is not even, then <math> x^2> </math> is not even, given that an odd number times an odd number must be odd (that is, not even). | + | Let <math> P </math> and <math> Q </math> be propositions such that <math> P \implies Q </math>. Then, the contrapositive of the conditional statement "<math> P \implies Q </math>" (read as "if P, then Q" or "P implies Q") is "<math> \neg Q \implies \neg P </math> (read as "if not Q, then not P" or "not Q implies not P"). The contrapositive is logically equivalent to the original conditional; that is, a conditional and its contrapositive always have the same truth values. For example, the contrapositive of "if <math> x > 0 </math>, then <math> x </math> is positive" is "if <math> x </math> is NOT positive, then <math> x \leq 0 </math>". These two statements are logically equivalent, and are both true. Sometimes, proving the contrapositive of a conditional is easier than proving the conditional itself. |
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| + | For example: Let x be an integer. If <math> x^2 </math> is even, then x is even. While we can attempt to prove this conditional statement directly, it is easier to show that if x is not even, then <math> x^2> </math> is not even, given that an odd number times an odd number must be odd (that is, not even). | ||
==Resources== | ==Resources== | ||
| + | * [https://en.wikipedia.org/wiki/Proof_by_contrapositive Proof by Contrapositive], Wikipedia | ||
| + | * [https://math.dartmouth.edu/~m22x17/misc/LaLonde2012_proof_by_contrapositive.pdf Proof by Contrapositive], Dartmouth University | ||
| + | * [http://zimmer.csufresno.edu/~larryc/proofs/proofs.contrapositive.html Proof by Contrapositive], Fresno State University | ||
Latest revision as of 10:49, 1 October 2021
Let and be propositions such that . Then, the contrapositive of the conditional statement "" (read as "if P, then Q" or "P implies Q") is " (read as "if not Q, then not P" or "not Q implies not P"). The contrapositive is logically equivalent to the original conditional; that is, a conditional and its contrapositive always have the same truth values. For example, the contrapositive of "if , then is positive" is "if is NOT positive, then ". These two statements are logically equivalent, and are both true. Sometimes, proving the contrapositive of a conditional is easier than proving the conditional itself.
For example: Let x be an integer. If is even, then x is even. While we can attempt to prove this conditional statement directly, it is easier to show that if x is not even, then is not even, given that an odd number times an odd number must be odd (that is, not even).
Resources
- Proof by Contrapositive, Wikipedia
- Proof by Contrapositive, Dartmouth University
- Proof by Contrapositive, Fresno State University
