Lecture 04 Slides
.pdf
School
University of Illinois, Urbana Champaign**We aren't endorsed by this school
Course
MATH 346
Subject
Mathematics
Date
Dec 21, 2024
Pages
10
Uploaded by SuperHumanPencil9860
Introduction to methods of proofc�S. Ahlgren 2020
Goals1.Gain familiarity with three basic methods of proof: Direct proof, proof bycontrapositive and proof by contradiction2.Apply these methods to prove theorems.The numbered examples in green(Ex 1., Ex. 2. etc.)are for you to try; please pause thevideo and work on the question before viewing the solution.You can skip directly to each solution using the “chapters" tab on the left.
Some sample theoremsThm 1.The sum of any three consecutiveintegers is divisible by3Thm 2.Ifnis an integer and5n+7is odd, thennis even.Thm 3.There is no largest integer.Thm 4.3x3−2x2−7has no integer roots.
Basic proof methods: Direct, Contrapositive, ContradictionTo proveP=⇒Q:Direct:Contrapositive:Contradiction:
ExampleThm 1.The sum of any three consecutive integers is divisible by3.
ExampleThm 2.Ifnis an integer and5n+7is odd, thennis even.
ExampleThm 3.There is no largest integer.
ExampleThm 4.3x3−2x2−7has no integer roots.
Ex 1.Prove the following statements about integersn.1.Ifn2is even thennis even.2.Ifn2is odd thennis odd.3.n2is even if and only ifnis even.
Goals1.Gain familiarity with three basic methods of proof: Direct proof, proof bycontrapositive and proof by contradiction2.Apply these methods to prove theorems.