2024年5月1日发(作者：)

PROOF,CONTRAPOSITIVE,CONTRADICTION

MATHCIRCLE(BEGINNERS)04/29/2012

P=⇒Qmeans“IfPthenQ.”ForexamplewhenPisthestatement“Annikadrives

acar”,andQisthestatement“Annikahasadriver’slicense”,thenP=⇒Qis

thestatement“IfAnnikadrivesacar,thenshehasadriver’slicense.”Thiskindof

statementiscalledanimplication,oraconditionalstatement.

Wewilluse¬Ptorepresentthestatement“Pisfalse.”WithPandQasabove,¬P

representsthestatement“Annikadoesnothaveadriver’slicense.”

Foreachofthefollowingstatements,writedownthestatementwhichisitslogical

negation:

(a)“ClintistheworstMathCircleinstructor.”

(b)“Thereexistsanevennumbergreaterthanorequalto4whichisnotthesumof

twoprimenumbers.”

(c)“Allswansarewhite.”

(d)“LondonisthecapitalofEngland,andSydneyisthecapitalofAustralia.”

(e)“Thesearen’tthedroidsyou’relookingfor.”

c

2008-2011OlgaRadko/LosAngelesMathCircle/UCLADepartmentofMathematics

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