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CPSC121–TutorialWeek8 1. Translatethefollowingtheoremsintopredicatelogic,andthenprovethem.Youmayutilizethe followingpredicates:Even(x),Odd(x),Square(x),Prime(x) a. Theproductofanythreeconsecutiveintegersisdivisibleby6. b. Iftwonumbersareperfectsquaresthentheirproductisaperfectsquare. c. Foreverydistinctpairofrealnumbers,thereisanotherrealnumberbetweenthem. d. Thereisexactlyoneevenprimenumber ! # !%# 2. Prove that for every distinct pair of positive rational numbers and , the number lies ! # " ! $ # "%$ between and .Hint,withoutlossofgenerality,youcanassume > ,andthata,b,c,andd " $ " $ areallpositive.Ingeneral,agoodwaytoprovethatxisgreaterthanyistoshowthat𝑥 − 𝑦 = 0 3. Extraproblem-TranslatethefollowingsentenceintoEnglish,giventhefollowingdefinitions: i. P domainofallpeople ii. K(x,y) personxknowspersony iii. L(x,y) personxislivingincountryy “EverypersonlivinginCanadaknowssomebodywhoknowsnobodylivinginBrazil”