# When is the empty function injective? surjective? bijective?

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.

• If "idea" is to mean more than "guess", why not give us your reasoning? Mar 17 '20 at 11:16
• Hint: How many elements of the empty set are mapped to a given element of $X$? Mar 17 '20 at 11:17
• I tried to reason this. Is this O.K.? Mar 17 '20 at 11:28
• Yov've got the definition of injective wrong! Mar 17 '20 at 11:44
• @PeterSmith Thanks, I edited. Is tihs O.K.? Mar 17 '20 at 11:48

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.