Let S = { 1 , 2 , 3 , 4 }. Let f be the sets of all functions from S to S.Now, is this statement correct? ∀ f ∈ F, ∃ g ∈ F so that ( f ∘ g ) ( 1 ) = 2This is false because suppose the only output f ( x ) can produce is f ( x ) = 3 and f ( x ) = 4Now, there is no possibility of ( f ∘ g ) ( 1 ) = 2 because 2 isn't an output of f.Is that correct to say?

Question

Let   S  =  {  1  ,  2  ,  3  ,  4  }. Let   f be the sets of all functions from   S to   S.Now, is this statement correct?  ∀  f  ∈  F,   ∃  g  ∈  F so that   (  f  ∘  g  )  (  1  )  =  2This is false because suppose the only output   f  (  x  ) can produce is  f  (  x  )  =  3 and   f  (  x  )  =  4Now, there is no possibility of   (  f  ∘  g  )  (  1  )  =  2 because   2 isn&#039;t an output of   f.Is that correct to say?

ivice7u · Accepted Answer

Yes, this is correct. Perhaps easier is to consider f(x)=1 for all x, but the idea is the same.

Tony Burgess · Accepted Answer

Yes. This could be tidied up a bit, but it is the correct idea. I would phrase it as &quot;Consider the constant function   f     :     S  →  S such that   f  (  x  )  =  4 for all   x. Then for any   g  ∈  F recognize that   g  (  x  )  ∈  S for each   x and so for each   x we have   (  f  ∘  g  )  (  x  )  =  f  (  g  (  x  )  )  =  4  ≠  2, providing a counterexample to the claim&quot;

Let S={1,2,3,4}. Let f be the sets of all functions from S to S. Now, is this statement correct? ∀f∈F, ∃g∈F so that (f∘g)(1)=2

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