Question 1179552
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Prove that if f : A → B is a function from A to B, then f ◦ iA = f and iB ◦ f = f.
(iA: inverse of A).
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Then  (f o iA)  is identical map of the image of  A  (which is part of  B)  to  B.


It is  NOT  TRUE  that   (f o iA) = f,   as you write in your post.




Also,  then (iA o f)  is identical map of  A  to  A.


It is  NOT  TRUE  that   (iB o f) = f,  as you write in your post.




Also, notice that  " iB "  is not defined in this problem and in this post,  at all.  


" iB  "  is totally fictitious subject,  irrelevant to this problem.



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Where,  from which source,  did you retrieve these incorrect statements ?


Do you create / compose / invent them on your own ?



If you retrieve it from some source,  then keep in mind that this source is  UNTRUSTED.




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Notice that in this my post,  I make not only my work as a tutor.


I make your part of work,  too  (also),  trying to formulate the problem  CORRECTLY.



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And the final note.


Everything what I wrote above, &nbsp;is written under the assumption that  &nbsp;&nbsp;iA &nbsp;is &nbsp;the inverse of &nbsp;A,  &nbsp;<U>as it is stated in your post</U>.



I think that it is the key error in your post.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If to assume that &nbsp;iA &nbsp;is &nbsp;IDENTICAL &nbsp;map of &nbsp;A &nbsp;to itself,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and if to assume that &nbsp;iB &nbsp;is &nbsp;IDENTICAL &nbsp;map of &nbsp;B &nbsp;to itself,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;then all statements in your post become &nbsp;a) &nbsp;&nbsp;correct and &nbsp;b) &nbsp;&nbsp;self-evident.