1
$\begingroup$

Denote $E$ and $F$ to be sets and $f:E\longrightarrow F$ a function and $\Gamma=\{(x,y)\in E\times F| y=f(x)\}$. I read that two functions are equal if and only if the have the same $E$, the same $F$, and the same $\Gamma$. I thought that equality of two functions does not depend on the set $F$, we can just take any set containing the image of $f$ and we still have the same function. Can we say that two functions are equal if and only if they have the same domain and the same graph (without mentionning the set $F$)?

$\endgroup$
2
$\begingroup$

It depends on convention. Some just take a function to be a set of ordered pairs such that no two distinct ordered pairs have the same first component. In that regard, a function is its graph, to a certain extent, so functions are indeed equal if and only if they have the same (domain and) graph.

For some, the codomain is a characteristic of the function itself--so, for example, surjectivity becomes a property inherent to a given function (as injectivity always is)--and in such cases, all three must match to have equality.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.