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Proof by contradiction is similar to refutation by contradiction, [4] [5] also known as proof of negation, which states that ¬P is proved as follows: The proposition to be proved is ¬P. Assume P. Derive falsehood. Conclude ¬P. In contrast, proof by contradiction proceeds as follows: The proposition to be proved is P. Assume ¬P. Derive ...
This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form. [4] The steps are as follows. All sentences in the knowledge base and the negation of the sentence to be proved (the conjecture) are conjunctively ...
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a 2 + b 2. Prove that a 2 + b 2 / ab + 1 is a perfect square. Let a 2 + b 2 / ab + 1 = q and fix the value of q. If q = 1, q is a perfect square as desired. If q = 2, then (a-b) 2 = 2 and there is no integral solution ...
Many proofs by contradiction use negation introduction as reasoning scheme: to prove ¬P, assume for contradiction P, then derive from it two contradictory inferences Q and ¬Q. Since the latter contradiction renders P impossible, ¬ P must hold.
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. [1] [2] More specifically, in trying to prove a proposition P, one first assumes ...
Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved [2] because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge.
For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd": (i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.