【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.

1回答
2020-02-08 18:38
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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.

2020-02-08 18:40:06

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