2 proof by contradiction Let r be a proposition.A proof of r by contradiction consists ofproving that not(r) implies a contradiction,thus concluding that not(r) is false,which implies that r is true.
3 proof by contradiction In particular if r isif p then qthen not(r) is logically equivalentto p AND (not(q)).This can be verified by constructinga truth table.
4 proof by contradiction So a true propositionif p then qmay be proved by contradiction as follows:Assume that p is true and q is false,and show that this assumption impliesa contradiction.
5 proof by contradiction One way of provingthat the assumption thatp is true and q is false impliesa contradiction is proving that thisassumption implies that p is false.Since the same assumption alsoimplies that p is true, we conclude thatthe assumption implies that p is true and pis false, which is a contradiction.
6 proof by contradiction EXAMPLE:Prove that the sum of an even integerand a non-even integer is non-even.(Note: a non-even integer is an integerthat is not even.)
7 proof by contradiction We have to prove that for every even integera and every non-even integer b, a+bis non-even.This is the same as proving thatFor all integers a,b, if [a is even andb is non-even] then [a+b is non-even].
8 proof by contradiction We prove that by contradiction.Assume that[a is even and b is non-even],and that [a+b is even]. So for someintegers m,n, a=2m and a+b=2n.Since b=(a+b)-a, b=2n-2m=2(n-m).We conclude that b is even. This leadsto a contradiction, since we assumed thatb is non-even.