# Is the empty function constant

Consider this definition of a constant function found in Lee:

Definition. A map $f: X \to Y$ is called a constant map if there is some element $c \in Y$ such that $f(x) = c$ for every $x \in X$.

If $X = \varnothing$, then $f = \varnothing \subseteq X \times Y$. Taking an arbitrary $c \in Y$, we have $f(x) = c$. Therefore the empty function is constant. Am I right? What is the case if $Y = \emptyset$? This is rather confusing, but I would say the function $f: X \to \varnothing$ is non-constant.

• See here, and this MSE-question. – Dietrich Burde Oct 1 '16 at 19:06
• @DietrichBurde I've already seen this. Thanks. I've just wanted to study this particular definition in detail. – TheGeekGreek Oct 1 '16 at 19:07

Indeed, the empty graph is (by this definition) a constant map in a vacuous sense, so long as $Y$ is non-empty.

In particular: if $f:\emptyset \to Y$ is given by $f = \emptyset$, then we can fix an element $c \in Y$ and state for every $x \in \emptyset$, $f(x) = c$. It so happens that there aren't any elements of $\emptyset$, so the statement stands.

Notably, $Y$ cannot be empty. In particular, there must exist an element of $Y$ for the definition to apply.

Note also that the only function that maps to the empty set is $f:\emptyset \to \emptyset$ given by $f = \emptyset$. If $X$ is non-empty, then no relation $f:X \to \emptyset$ can be defined over its entire domain. That is, $f(x)$ is not defined for any $x \in X$, because there are no pairs $(x,y) \in f$. So, it's not that $f: X \to \emptyset$ fails to be a constant function, it fails to be a function at all.

• Whether or not $\emptyset\to\emptyset$ is constant depends on the wording of the deifnition of "constant function". The OP uses $\exists c\in Y\colon\forall x\in X\colon f(x)=c$. If one uses $\forall x_1,x_2\in X\colon f(x_1)=f(x_2)$ instead, then $\emptyset\to\emptyset$ is constant (which seems to be more natural to me - for a non-constant function I'd much love to assume that it takes at least two values). Also, I am unhappy with the effect that constantness depends on the codomain chosen. – Hagen von Eitzen Oct 1 '16 at 19:14
• @HagenvonEitzen I agree, though the OP is interested in this particular definition – Omnomnomnom Oct 1 '16 at 19:16
• @HagenvonEitzen Oh very nice! That solves my problem why I am stating this question. I will just use the definition you've mentioned. Perfect. But I will accept this answer since it solves the question stated here. +1 – TheGeekGreek Oct 1 '16 at 19:17

Yes, the empty function $f \colon \emptyset \to Y$ is constant, for the very reason that you stated. On the other hand, there is no function $f \colon X \to \emptyset$ unless $X= \emptyset$ (and if $X = \emptyset$, then it is constant - see above). The reason that, for $X \neq \emptyset$, there is no function $f \colon X \to \emptyset$ lies in the definition of a function. Since $X \neq \emptyset$, there is some $x \in X$ and - by the definition of a function - there must then be a unique $y \in \emptyset$ such that $(x,y) \in f$. But this is absurd.

• Thanks. But if $Y = \varnothing$, then I cannot choose an element $c \in Y$ and therefore $f: \emptyset \to \emptyset$ is non-constant? – TheGeekGreek Oct 1 '16 at 19:14
• You just don't have to pick an element at all. For all $x \in X$ there must be a unique $y \in Y$ such that $(x,y) \in f$. If there is no $x \in X$, then you don't have to do anything (; – Stefan Mesken Oct 1 '16 at 19:16
• @TheGeekGreek That depends on the exact wording of the definition (cf. my comment to the other answer) – Hagen von Eitzen Oct 1 '16 at 19:17
• @HagenvonEitzen Yep, I read the comment. Good point. – Stefan Mesken Oct 1 '16 at 19:17
• @Stefan Thanks for your answer. I think the definition Hagen mentioned is more appropriate. But good answer. +1 – TheGeekGreek Oct 1 '16 at 19:20