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Can a function have multiple codomains? The definition of codomain of a function is that it is a superset of the range of the function. But I have doubt whether the definition means that a codomain is

  1. any superset of range of the function.
  2. a superset of range, defined by the author of a problem.

From 1 it is clear that there can be infinite codomains of a function. For example suppose we have $f=\{(1,1), \ (2,3)\}$ then in sense 1 the possible codomains can be $c_1=\{1,3,2\}$ $c_2=\{1,4,2\}$. But I never encountered mention of multiple codomains in any book and hence I think the standard definition of codomain is the second one. And hence a problem must specify its codomain, like by using the notation $f:A\rightarrow B$ where $A=\{1,2\}$ and $B=\{1,5,2\}$ and hence $f$ has only one codomain, in sense 2.

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  • $\begingroup$ "codomain of a function is ... a superset of the range" $\;-\;$ That's a rather odd way to say that the range is a subset of the codomain. Anyway, that is not a valid definition of the codomain, and the page you linked does not use that as a definition. $\endgroup$ Commented Feb 22, 2022 at 6:47
  • $\begingroup$ Some authors define a function to be an ordered triple $(f,A,B)$ where $A$ and $B$ are sets and $f$ is a subset of $A \times B$ which satisfies certain properties. The set $B$ is called the codomain of the function. For those authors, a function has a unique codomain which is specified when the function is introduced. Other authors define functions in a way that doesn't specify a particular codomain, and those authors might avoid using the term "codomain" entirely. $\endgroup$ Commented Feb 22, 2022 at 6:52
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    $\begingroup$ See also this post as well as this one $\endgroup$ Commented Feb 22, 2022 at 6:56

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