Set Theory – When is the Empty Function Injective, Surjective, and Bijective?

Question

Let us think about the empty function $f:\emptyset\rightarrow X$ ($X$ is an arbitrary set.) .

My idea is $f$ is always injective. Iff $X=\emptyset$, $f$ is surjective (so bijective).

(Reasoning)

Definition of injective is :$x\neq x'\rightarrow f(x)\neq f(x')$. The empty set has no element, so $x\neq x'\rightarrow f(x)\neq f(x')$ is always true.

Definition of surjective is : $\forall y\in Y$, there exists $x\in X$ such that $f(x)=y$.

Iff Y is the empty set,there is no element of $Y$, so $\forall y\in Y$, there exists $x\in X$ such that $f(x)=y$ is true.

Answer 1

You are correct. Since $\forall x, y \in \emptyset f(x) = f(y) \Longrightarrow x = y$ is vacuously true, $f$ is always injective.

Now, let's check surjectivity (namely $\forall x \in X\ \exists y \in \emptyset, f(y) = x$). $f$ is:

• not surjective if $X$ contains any element $x$ since $\exists y \in \emptyset, f(y) = x$ is false.
• surjective if $X = \emptyset$. Again, the sentence $\forall x \in \emptyset \ \exists y \in \emptyset, f(y) = x$ is vacuously true.