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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?

Filippo
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  • @JoséCarlosSantos Thank you for the link, based on the title I would say yes, but I will read the question and the answers to give a definite answer. – Filippo Jul 24 '22 at 12:03
  • @JoséCarlosSantos Okay, I have read the question, the answer and all the comments. In summary, I would say that there was some disagreement and the issue can not be considered entirely resolved (the only answer that was given has not been accepted). IMO it would be worth keeping my question in order to see if more people participate in the discussion (actually this already happened, GEdgar did not participate in the other discussion). Is that okay? – Filippo Jul 24 '22 at 12:49
  • I think that it is a duplicate. Let us see what other users think. – José Carlos Santos Jul 24 '22 at 13:07
  • I agree, it certainly is a duplicate. But IMO the issue is not solved yet, so I would like to hear more opinions and get more information. – Filippo Jul 24 '22 at 14:55

1 Answers1

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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).

GEdgar
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  • Okay, so if we are "pedantic", then the answer to my question is NO. On the other hand, I assume that your answer to "Is the graph of a function from X to Y also the graph of a function from X to Z?" is YES. I am wondering if the graph represents what I referred to as an "assigment of elements in Y to elements in X"... – Filippo Jul 24 '22 at 13:51
  • I was hoping to hear your opinion (since this was part of my question), but I learned a new vocabulary (astriction), so +1. – Filippo Jul 25 '22 at 06:07