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A function is described in terms of its domain, its range (image) and its codomain. The first two are well defined sets but I found the codomain of a function as a bit ambiguous as according to me a function can have infinite codomains, as codomain is merely defined as a set containing the range of the function, and their can be many such sets in most cases.

Then my question is that how we can define codomain of a function uniquely and what is its use ?

To specify, I also found similar questions on the site but I could not be satisfied and want someone to help me in this by explaining this in detail as this is really confusing.

Thanks for help.

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What's going on here is that there are two different conventions about what "function" means.

I imagine you're focused on the graph of a function — the set of all points $(x, f(x))$.

  • In one convention, the graph of the function is sufficient to determine the function. By this convention, functions don't have well-defined codomains.
  • In the other convention, functions do have well-defined codomains, which means the graph of the function is not sufficient to determine the function.

One of the ways to encode the notion of function by the second convention is as a triple $(X, Y, \Gamma)$ where $X$ is the domain, $Y$ is the codomain, and $\Gamma$ is the graph.

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  • $\begingroup$ But can't there be many codomains in the second definition also if we take into account the definition of codomain as a set containing the set of values taken by the function (range). If this is not the case then how codomain is defined and how it is used. Thanks for your help but I'm finding it incomplete as it has not answered these questions. $\endgroup$ – Abhinav Dhawan Apr 12 '17 at 1:48
  • $\begingroup$ @AbhinavDhawan: If, for example, you use the representation of functions I described in my post, then the codomain of $(X,Y,\Gamma)$ is $Y$. $\endgroup$ – user14972 Apr 12 '17 at 13:45

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