【数学简单证明英语LetA,B,C,andDbenonemp-查字典问答网

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　　【数学简单证明英语LetA,B,C,andDbenonemptysets.ThenA×B=C×DifandonlyifA=CandB=D.(a)Provethisstatement.(b)Oneofthetwoimplicationsdoesnotrequirethesetstobenonempty.Whichone?(c)Ifwedonotrequirethesetsto】

　　数学简单证明英语

　　LetA,B,C,andDbenonemptysets.ThenA×B=C×DifandonlyifA=CandB=D.

　　(a)Provethisstatement.

　　(b)Oneofthetwoimplicationsdoesnotrequirethesetstobenonempty.Whichone?

　　(c)Ifwedonotrequirethesetstobononempty,thenthestatementisfalse.GiveexamplesofsetsA,B,CandDtoshowthenecessityoftheassumptionthatthesetsbenonempty.

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2020-02-0818:38

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况百杰

　　(a)ifA=CandB=D,ofcourseAxB=CxD.

　　ifAxB=CxDandA,B,C,Dnon-empty.IfAisnotequaltoC,let'ssayAhasanelementathatdoesnotbelongtoC.SinceBandDarenon-empty,theremustbeapairwithelementainAxC,butthatpairdoesnotbelongtoCxDsinceelementadoesnotbelongtoCinthefirstplace.SoAmustbeequaltoC.LikewiseBmustbeequaltoD.

　　(b)ifA=CandB=D,thenAxB=CxD.

　　(c)ifAisempty,AxBisempty,ifCisempty,CxDisalsoempty.SoBcandifferfromD,butAxBstillequalsCxD.