Difference between revisions of "Proofs:Contraposition"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 2: Line 2:
  
 
==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

Revision as of 09:59, 24 September 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