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I am new to the community. I am self-studying the real analysis from Analysis I by Terence Tao and I had a question related to function. I don't know the reason behind the statement.

A rather boring example of a function is the empty function f : ∅→ X from the empty set to a given set X. Since the empty set has no elements, we do not need to specify what f does to any input. Nevertheless, just as the empty set is a set, the empty function is a function, albeit not a particularly interesting one. Note that for each set X, there is only one function from ∅ to X, since Definition 3.3.8 asserts that all functions from ∅ to X are equal (why?).

How can I tackle this and any formal proofs can be provided? Thank you.

Asaf Karagila
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Chak On LEUNG
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    It is because its graph is $\varnothing,$ as a subset of $\varnothing\times X=\varnothing.$ – Anne Bauval Jul 18 '23 at 05:47
  • Does it mean that X is an empty set? – Chak On LEUNG Jul 18 '23 at 06:06
  • No, the argument holds for any codomain $X,$ not necessarily equal to the empty set. – Anne Bauval Jul 18 '23 at 06:10
  • That's vacuously true. Let f and g be any such functions. Now let x be an element of the empty set, therefore, f(x)=g(x). Since x was arbitrary we have that f=g by definition of equality of functions. I used that a false statement can imply any other statement (that's a vacuous truth) in this particular case x is in the empty set which is a false statement. – Bastián Jul 18 '23 at 06:23
  • The empty set can only be mapped to itself, that is why all such functions are the same. – Abezhiko Jul 18 '23 at 06:31
  • @Abezhiko That is insufficient; one also needs the fact that for each set $X$ there is a unique embedding of $\emptyset$ into $X$. – ancient mathematician Jul 18 '23 at 06:34

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