Hey [John](https://forum.azimuthproject.org/profile/17/John%20Baez),

> Puzzle (A). Exactly which functions have left inverses?

>

> Puzzle (B). Exactly which functions have right inverses?

We've chatted about this before but it's always good to review.

**Lemma (A)**. In ZF, a function is *injective* if and only if it has a *left* inverse.

**Lemma (B)**. In ZFC, a function is *surjective* if and only if it has a *right* inverse.

In fact, Lemma (B) is equivalent to the [axiom of choice](https://en.wikipedia.org/wiki/Axiom_of_choice).

I have to run, but I will try to swing by later and give proofs of **Lemmas (A)** and **(B)** if someone doesn't beat me to it.

> Puzzle (A). Exactly which functions have left inverses?

>

> Puzzle (B). Exactly which functions have right inverses?

We've chatted about this before but it's always good to review.

**Lemma (A)**. In ZF, a function is *injective* if and only if it has a *left* inverse.

**Lemma (B)**. In ZFC, a function is *surjective* if and only if it has a *right* inverse.

In fact, Lemma (B) is equivalent to the [axiom of choice](https://en.wikipedia.org/wiki/Axiom_of_choice).

I have to run, but I will try to swing by later and give proofs of **Lemmas (A)** and **(B)** if someone doesn't beat me to it.