For any set A , if f: A -> A is one-to-one the f must beonto. Solution Question

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Question Details: For any set A, if f: A-> A is one-to-one the f must be onto. THE FUNCTION IS GIVEN TO BE FROMA TO A , THAT IS DOMAIN AS WELL AS RANGE ARE THE SET A . THAT IS IF A IS [X1,X2,X3,........XN] THEN F[A] IS ALSO[X1,X2,X3,...XN]..POSSIBLY IN A DIFFERENT ORDER SO THAT F[XI] = YI , WHERE YI = SOME ELEMENTXJ , WHERE I AND J ARE FROM 1 TO N . LET X1,X2 BE ANY 2 ELEMENTS OF A AND Y1 AND Y2 THEIR IMAGESUNDER THE FUNCTION F[A]...THAT IS F[X1]=Y1 F[X2]=Y2 FUNCTION IS ONE TO ONE HENCE IF Y1=Y2 IT IMPLIES X1=X2....THATIS ALL ELEMENTS IN SET A...THAT IS ...X1,X2,....XN ARE ALL DISTINCTELEMENTS NOW LET US TAKE ANY ELEMENT Y' INF[A]. SO Y' = SOME XJ IN A AS ELABORATEDABOVE THE FUNCTION BEING ONE TO ONE , AS DISCUSSED ABOVE , IT SHALL HAVESOME ELEMENT XI SO THAT F[XI]=XJ=Y' SO FOR ANY ELEMENT Y' IN F[A] WE HAVE A PRE IMAGE XI IN A SO THE FUNCTION IS ON TO

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