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# Lecture_5 - MA1100 Lecture 5 Mathematical Proofs...

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1 MA1100 Lecture 5 Mathematical Proofs Definitions Axioms Direct Proof

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MA1100 Lecture 5 2 True Statements • Mathematical statements that are true are known as mathematical facts . • A proposition is a true mathematical statement that has a proof . • A proposition that is “ important ” is called a theorem . • A proposition that is used in the proof of a theorem is called a lemma . • A proposition that follows (easily) from a theorem is called a corollary . • There are true statements that do not have proofs e.g. definitions , axioms .
MA1100 Lecture 5 3 An Example If x is an even integer, then x 2 is divisible by 4. Proposition We want to give a proof for this (universal) conditional statement. We need to : know the definitions of even integers and divisibility use some basic facts about integers use algebraic manipulation

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MA1100 Lecture 5 4 Definition Examples (Mathematical) 1. Even and odd integers 2.n is divisible by m 3. Closure properties of a number system A (mathematical) definition gives the precise meaning of a word or phrase that represents some object, property or other concepts.
MA1100 Lecture 5 5 Even integers 2 6 -14 Example An integer a is an even integer if there exists an integer n such that a = 2n . Definition In general, an even integer can be written in the form 2n

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MA1100 Lecture 5 6 Odd integers 3 7 -13 Example An integer a is an odd integer if there exists an integer n such that a = 2n + 1 . Definition In general, an odd integer can be written in the form 2n + 1
MA1100 Lecture 5 7 Even and odd integers Why do we need definitions for even and odd integers? Is 0 an even integer? Not to identify specific even and odd integers To describe general even and odd integers To be used in the proof of statements involving even and odd integers

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MA1100 Lecture 5 8 Undefined Terms Some basic concepts remain undefined . • Numbers, integers, … • Addition, multiplication, … • Point, line, … Examples
MA1100 Lecture 5 9 Definition Form A definition is a universal conditional statement.

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