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 | * [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 | * [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 0 } , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } is positive" is "if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } is NOT positive, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq 0 } ". 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2> } 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
