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Proof Methods - Mathematical and Statistical Sciences

Contradiction Proof Example: The √2 is irrational. Pf: BWOC assume that √2 is rational. There exist integers p and q so that √2 = p/q. We may assume that the fraction is reduced, i.e. no integer divides both p and q. 2 = p2/q2 ⇒ 2q2 = p2, so p2 is even. Thus, p is even. Definition: A real number r is rational iff it can be written as r ...

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Linear Codes

www-math.ucdenver.edu

convert from the generator matrix to the parity-check matrix and vice-versa. This is easily done. Theorem 2: If G = [I k A] is the generator matrix (in standard form) for the [n,k]-code C, then H = [-AT I n-k] is the parity check matrix for C. Proof.

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Higher Geometry - Mathematical & Statistical Sciences

www-math.ucdenver.edu

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www-math.ucdenver.edu

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The Chinese Remainder Theorem - UC Denver

www-math.ucdenver.edu

by 3, and remainder 3 when divided by 7. We are looking for a number which satisfies the congruences, x ≡ 2 mod 3, x ≡ 3 mod 7, x ≡ 0 mod 2 and x ≡ 0 mod 5.

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Equivalence Relations - Mathematical and Statistical Sciences

www-math.ucdenver.edu

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Direct proof

people.cs.pitt.edu

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Solutions to Homework Set 3 (Solutions to Homework ...

math.okstate.edu

Proof.) Suppose a 6= [0], b 6= [0] and that ab = [0]. We aim to show that ax = [1] has no solution. We will use a proof by contradiction. Suppose c is a solution of ax = [1]. Then b = b1 = b(ac) = (ab)c = [0] c = 0 : But this contradicts our original hypothesis that b is a nonzero solution of ax = [0]. Hence, there can be no solution of ax = [1].

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cs.stanford.edu

In proof by contradiction, you assume your statement is not true, and then derive a con-tradiction. This is really a special case of proof by contrapositive (where your \if" is all of mathematics, and your \then" is the statement you are trying to prove). 2. 1.2 Proof by induction 1 PROOF TECHNIQUES

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2. Propositional Equivalences 2.1. Tautology/Contradiction ...

www.math.fsu.edu

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WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE …

www2.math.uconn.edu

is the negation of a contradiction. A typical contradiction is P^:P. Try to think yourself of some other examples. 2.3. Axioms. As it turns out, to prove something requires the knowl-edge of some previous truths. Logic just supplies the ways that we can deduce a statement from others, but we need some statements to begin with.

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PART I. THE REAL NUMBERS - UH

www.math.uh.edu

Proof: Suppose there exists an >0 such that the interval (u− , u] contains no points of S. Then s ≤ u − for all s ∈ S, which implies that u − is an upper bound for S which is less than u, a contradiction. Definition 5.:LetS ⊆ R be a set that is bounded below. A …

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Proof by Contradiction - Gordon College

www.math-cs.gordon.edu

Proof by Contradiction This is an example of proof by contradiction. To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false. Many of the statements we prove have the form P )Q which, when negated, has the form P )˘Q. Often proof by contradiction has the form ...

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