PROOF BY CONTRADICTION

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PROOF BY CONTRADICTION

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proof by contradiction

• Let r be a proposition.
• A proof of r by contradiction consists of
• proving that not(r) implies a contradiction,
• thus concluding that not(r) is false,
• which implies that r is true.

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proof by contradiction

• In particular if r is
• if p then q
• then not(r) is logically equivalent
• to p AND (not(q)).
• This can be verified by constructing
• a truth table.

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proof by contradiction

• So a true proposition
• if p then q
• may be proved by contradiction as follows
• Assume that p is true and q is false,
• and show that this assumption implies
• a contradiction.

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proof by contradiction

• One way of proving
• that the assumption that
• p is true and q is false implies
• a contradiction is proving that this
• assumption implies that p is false.
• Since the same assumption also
• implies that p is true, we conclude that
• the assumption implies that p is true and p
• is false, which is a contradiction.

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proof by contradiction

• EXAMPLE
• Prove that the sum of an even integer
• and a non-even integer is non-even.
• (Note a non-even integer is an integer
• that is not even.)

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proof by contradiction

• We have to prove that for every even integer
• a and every non-even integer b, ab
• is non-even.
• This is the same as proving that
• For all integers a,b, if a is even and
• b is non-even then ab is non-even.

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proof by contradiction

• We prove that by contradiction.
• Assume that
• a is even and b is non-even,
• and that ab is even. So for some
• integers m,n, a2m and ab2n.
• Since b(ab)-a, b2n-2m2(n-m).
• We conclude that b is even. This leads
• to a contradiction, since we assumed that
• b is non-even.