Direct proof
Proof 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.
Download Direct proof
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document:
Advertisement
Documents from same domain
CS 1652: Data Communication and Computer Networks …
people.cs.pitt.edufrom the application layer to the data-link layer. Concurrent with the lectures, you (in groups of two) will be building a functional TCP/IP stack and a small web server that will run on it.
Network, Computer, Communication, Data, Data communication and computer networks
Informal proofs - University of Pittsburgh
people.cs.pitt.eduInformal proofs Proving theorems in practice: • The steps of the proofs are not expressed in any formal language as e.g. propositional logic
Language, Informal, Proof, Formal, Formal language, Informal proofs
MIPS Floating Point Instructions
people.cs.pitt.edu11/9/2011 1 MIPS Floating Point Instructions CS/COE 447 Why Floating Point? • Sometimes need very small, or very large numbers? Non-integers?
Instructions, Points, Floating, Imps, Floating point, Mips floating point instructions
Foundations of Artificial Intelligence
people.cs.pitt.edu“The branch of computer science that is concerned with the automation of in-telligent behavior” (Luger+Stubblefield, 1993) Views of AI fall into four categories: Thinking humanly Thinking rationally Acting humanly Acting rationally Examining these, we will plump for acting rationally (sort of) AIMA Chapter 1 (after Russell and Norvig) 3
Intelligence, Foundations, Sciences, Artificial, Foundations of artificial intelligence
Propositional logic: Horn clauses
people.cs.pitt.edu• Horn form (Horn normal form) • Two inference rules that are sound and complete with respect to propositional symbols for KBs in the Horn normal form: – Resolution (positive unit resolution) – Modus ponens (A∨¬B) ∧(¬A∨¬C ∨D) Can be written also as: (B ⇒ A) ∧(( A ∧C) ⇒ D) CS 2740 Knowledge Representation M. Hauskrecht ...
Sequences and summations
people.cs.pitt.eduSequences and summations CS 441 Discrete mathematics for CS M ... Arithmetic progression Definition: An arithmetic progression is a sequence of the ... -1, 3, 7, 11, … 3 CS 441 Discrete mathematics for CS M. Hauskrecht Geometric progression Definition A geometric progression is a sequence of the form: a, ar, ar2, ..., ark, where a is the ...
Time Series: Autoregressive models AR, MA, ARMA, ARIMA
people.cs.pitt.eduGaussian White Noise {A particular useful white noise is Gaussian white noise, wherein the w ... -20 0 20 40 60 80 12/77. Time Series Analysis The procedure of using known data values to t a time series ... Measures of Dependence A complete description of a time series, observed as a
Introduction to Kernel Methods
people.cs.pitt.edu1 Introduction to Kernel Methods Dave Krebs CS 3750 Fall 2007 ... Paradigm for Pattern Analysis. Kernel Methods in Bioengineering, Signal and Image Processing. 2007. ... 9 Mercer’s Condition (continued) if and only if, for any g(x) such that is finite, then It can be ...
Analysis, Introduction, 1 introduction, Patterns, Pattern analysis
Probabilities: Expected value
people.cs.pitt.eduExpected value Investment problem: • You have 100 dollars and can invest into a stock. The returns are volatile and you may get either $120 with probability of 0.4, or $90 with probability 0.6. • What is the expected value of your investment? • M. Hauskrecht Expected value Investment problem: • You have 100 dollars and can invest into a ...
Sets and set operations - University of Pittsburgh
people.cs.pitt.edu• Ordered-n tuples are used to represent an ordered collection. Definition: An ordered n-tuple (x1, x2, ..., xN) is the ordered collection that has x1 as its first element, x2 as its second element, ..., and xN as its N-th element, N 2. Example: • Coordinates of a point in the 2-D plane (12, 16) x y
Related documents
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 ...
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 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
Proof, Contradictions, Proof by contradiction, Con tradiction, Tradiction
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].
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 ...
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 …
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.