What is the difference between a proof by counterexample and a proof by contradi

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Proof by contradiction means to prove A=>B start with the opposite conjecture A=>~B and arrive at a contradition I think that should read "start with the conjecture that A does not imply B" rather than "start with the conjecture that A implies Not_B". More generally proof by contradiction doesn't need to be set out in the form of "A implies B" etc, but rather it just requires that we start with a conjecture "C" that we wish to prove and then show that if C is false then that leads to a contradiction. If the conjecture "C" that we begin with is that "A implies B" then assuming C to be false is equivalent to assumming that "A does not imply B" rather than assumming that "A implies Not_B". In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction. Since by the law of bivalence a proposition must be either true or false, and its falsity has been shown impossible, the proposition must be true.[citation needed] In other words, to prove by contradiction that P, show that or its equivalent . Then, since implies a contradiction, conclude P. Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem. It is a particular kind of the more general form of argument known as reductio ad absurdum.

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