Tao Analysis I - 3.3.3

Exercise 3.3.3

When is the empty function into a given set X injective? surjective? bijective?

Let's remind ourselves that an empty function $f: \emptyset \mapsto X$ maps 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. Note that each and every set $X$ has only one empty function from $\emptyset$ to that $X$.

When is the empty function injective?

Any function must map an input to only one output, but an injective function also requires that different inputs don't map to the same output. This is Definition 3.3.17.

$$x \neq x' \implies f(x) \neq f(x')$$

Since the empty function has no elements, the condition for an empty function $f: \emptyset \mapsto X$ to be injective is vacuously true. $\square$

When is the empty function surjective?

A function $f: A \mapsto X$ is surjective if every element $x$ of the codomain $X$ has an $a$ in the domain $A$ such that $f(a)=x$. This is Definition 3.3.20.

$$x \neq x' \implies f(x) \neq f(x')$$

Since the domain $A=\emptyset$ of an empty function has no elements, then the definition can't apply. The only exception is when the codomain is empty $X=\emptyset$, in which case the defining condition is vacuously true.

An empty function is surjective if the codomain $X$ is the empty set $\emptyset$. $\square$.

When is the empty function bijective?

A function is bijective if it is both injective and surjective. A bijective function is also called invertible. This is from Definition 3.3.23.

Given the above discussion, an empty function is always injective but only surective when the codomain is the emptyset. 

Thus, the empty function is bijective, invertible, when the codomain is the empty set $X=\emptyset$. $\square$