Proof By Contradiction
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Direct proof
people.cs.pitt.eduProof by contradiction Direct proof In a direct proof , we prove p →q by showing that if p is true , then q must necessarily be true Example: Prove that if n is an odd integer, then n 2 is an odd integer. Proof: Assume that n is odd. That is n = (2k + 1) for some integer k.
Solutions to Homework Set 3 (Solutions to Homework ...
math.okstate.eduProof.) 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].
Proof Techniques - Stanford University Computer Science
cs.stanford.eduIn 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
2. Propositional Equivalences 2.1. Tautology/Contradiction ...
www.math.fsu.eduAn alternative proof is obtained by excluding all possible ways in which the propositions may fail to be equivalent. Here is another example. Example 2.3.2. Show :(p!q) is equivalent to p^:q. ... ,F^q Contradiction,F Domination Law and Commutative Law Example 2.5.2. Find a simple form for the negation of the proposition \If the sun is shining ...
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE …
www2.math.uconn.eduis 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.
PART I. THE REAL NUMBERS - UH
www.math.uh.eduProof: 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 …
Proof by Contradiction - Gordon College
www.math-cs.gordon.eduProof 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 ...
Proof Methods - Mathematical and Statistical Sciences
www-math.ucdenver.eduContradiction 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 ...