If $Y\subset Z$, is a function from $X$ to $Y$ also a function from $X$ to $Z$?

Question

If $Y\subset Z$, is a function from $X$ to $Y$ also a function from $X$ to $Z$?

I think both possible answers are plausible:

• One the one hand, a function $f$ from $X$ to $Y$ is often called an assignment of elements in $Y$ to elements in $X$. By this definition, the answer to my question is clearly YES.
• If we say that each function has a domain and codomain associated to it (i.e. they are part of the data), then the answer is clearly NO.

I guess this is a question about definitions and conventions and I would like to hear your thoughts. I have the impression that I am considering two different concepts, one of them being functions. What is the other one?

Answer 1

Yes, sometimes we do this. It depends on the setting. Certainly do it only if it it not confusing to the reader.

To be pedantic, we may say that the function $X \to Z$ is an "astriction" of the function $X \to Y$. It has the same graph, but larger codomain. This is a counterpart notion to the more common "restriction", where we change the domain (and therefore also change the graph).