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MATH135 Fall 2013

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University of Waterloo
MATH 135
Ross Willard

Return to University Notes indexMATH 135Mathematical proofsSection 004InstructorName Ross Willard Office MC5058Email rdwillaruwaterlooca rosswillarduwaterloocaOffice Hours Tuesdays 900AM1000AMFridays 100PM200PM in MC 5058Tutorials Tuesdays 430PM720PM Dropin to Biology 1 Room 2719913Show xx1xxxxxxx1xxx1xxx1xxxThis proof is invalidProofsAvoid starting with thing trying to be proved assuming the thing is trueUse english sentences to give details2Prove that 2xx0x2xLet x represent a real number start by declaring assumptions2Assume 2xx2Then 2x0xThen xx20This implies that eitherandor thatand x0x2x0x2Ifandthen x0x2x2Ifandthen x0x2x02In conclusionandif x0x22xxCheck the converseAssumethenand x0xx02x02Therefore 2xxAssumethen x2xx2x2Therefore 2xx2In conclusionandif and only if x0x22xx1x21Showfor all x2x0xLetrepresent a positive real numberx2Certainlyis truex102So 2x10x22Thusand is equivalent to 2x12x02xx12xx12Since is equivalent tox012xxx2x1Hencewhich was to be provedx2xIn handwritingdouble right arrow is considered an acceptable replacement for Therefore Hence Then ThusA proposition is a true statement proved with a valid argumentA theorem is an important or significant propositionA lemma is a helper proposition used in the proof of a theoremA corollary is a proposition that follows almost immediately from a theoremAn axiom is a statement is assumed to be true no proof needed Propositions are derived from thesewip Read textbook ch 1 2 4 except section 24 5 6 7 8 13SummaryWritten solutions are primarily done in English Proofs start with relevant assumptions or known true statements and endswith the thing to be proved1191322Show xyxy4xyLetandrepresent real numbersxy222222Clearly xyxy2xy2xyxyxy22222222Clearly 2xy2xy2xy2xyxyxyxyxy2222Clearly 2xy2xy2xy2xyxyxy22Thusby transitivity of equalityxyxy4xyIn the interests of style multiple equalities can be written thusLetandrepresent real numbersxy222222Clearly xyxy4xy2xy2xyxyxy222222222xy2xy2xy2xyxyxyxyxy22222xy2xy2xy2xyxyxy22Thusby transitivity of equalityxyxy4xyheremeans which in turnOnly statements can imply other statementsStatements are sentences that can be true or false They are complete sentences with subjects and verbs true statement subject verb subject verb subject2242The equationhas 2 distinct real roots true statement subject verb subject10x context required open sentencex0Compound statements are composed of several component statements
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