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Quick questions about equality of functions!

  1. Jan 7, 2009 #1
    Well, i was reading a book lately about functions, and when it came to define the equality of two functions it defined something like this:

    Let f:A->B and g:C->D be two functions.

    We say that these two functions are equal if:

    2.B=D and
    3.f(x)=g(x) for all x in A=C.

    I guess i have always overlooked it, but is 2. a little bit redundant. I mean, would a more precise statement be to say that if: ran{f}=ran{g}, rather than in terms of Codomains of these functions?

    The reason i say this is that, for example:

    Let: f:N-->R be a function from Naturals to Reals defined as follows: f(n)=n+1

    and, let g:N-->Z be a function from Naturals to integers defined also as: g(n)=n+1

    From here we see that their domains are the same, the ranges are the same and also f(n)=g(n) for every n in the domain. Can we say from here that these two functions are the same, or not? I would say yes, but maybe i am overlooking something.

  2. jcsd
  3. Jan 7, 2009 #2
    By the definition given in your book, then no they aren't equal and that is the ordinary definition. I have never seen them defined in terms of ranges instead of codomains. Properties such as surjectiveness would stop making sense. Consider the floor function; with the real numbers as the codomain it would not be surjective, but with the integers as the codomain it would be surjective, but by your definition they are equal so functions can be both surjective and not surjective.
  4. Jan 7, 2009 #3
    Well, yeah that makes sense.

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