# The empty set is the only set which satisfies the universal property of the empty set

Here, we will prove that the only set which satis­fies the uni­ver­sal prop­erty of the empty set is the empty set it­self. This will tell us that defin­ing the empty set by this uni­ver­sal prop­erty is ac­tu­ally a co­her­ent thing to do, be­cause it’s not am­bigu­ous as a defi­ni­tion.

There are three ways to prove this fact: one way looks at the ob­jects them­selves, one way takes a more maps-ori­ented ap­proach, and one way is sort of a mix­ture of the two. All of the proofs are en­light­en­ing in differ­ent ways.

Re­call first that the uni­ver­sal prop­erty of the empty set is as fol­lows:

The empty set is the unique set $$X$$ such that for ev­ery set $$A$$, there is a unique func­tion from $$X$$ to $$A$$. (To bring this prop­erty in line with our usual defi­ni­tion, we de­note that unique set $$X$$ by the sym­bol $$\emptyset$$.)

# The “ob­jects” way

Sup­pose we have a set $$X$$ which is not empty. Then it has an el­e­ment, $$x$$ say. Now, con­sider maps from $$X$$ to $$\{ 1, 2 \}$$.

We will show that there can­not be a unique func­tion from $$X$$ to $$\{ 1, 2 \}$$. In­deed, sup­pose $$f: X \to \{ 1, 2 \}$$. Then $$f(x) = 1$$ or $$f(x) = 2$$. But we can now define a new func­tion $$g: X \to \{1,2\}$$ which is given by set­ting $$g(x)$$ to be the other one of $$1$$ or $$2$$ to $$f(x)$$, and by let­ting $$g(y) = f(y)$$ for all $$y \not = x$$.

This shows that the uni­ver­sal prop­erty of the empty set fails for $$X$$: we have shown that there is no unique func­tion from $$X$$ to the spe­cific set $$\{1,2\}$$.

# The “maps” ways

We’ll ap­proach this in a slightly sneaky way: we will show that if two sets have the uni­ver­sal prop­erty, then there is a bi­jec­tion be­tween them. noteThe most use­ful way to think of “bi­jec­tion” in this con­text is “func­tion with an in­verse”. Once we have this fact, we’re in­stantly done: the only set which bi­jects with $$\emptyset$$ is $$\emptyset$$ it­self.

Sup­pose we have two sets, $$\emptyset$$ and $$X$$, both of which have the uni­ver­sal prop­erty of the empty set. Then, in par­tic­u­lar (us­ing the UP of $$\emptyset$$) there is a unique map $$f: \emptyset \to X$$, and (us­ing the UP of $$X$$) there is a unique map $$g: X \to \emptyset$$. Also there is a unique map $$\mathrm{id}: \emptyset \to \emptyset$$. noteWe use “id” for “iden­tity”, be­cause as well as be­ing the empty func­tion, it hap­pens to be the iden­tity on $$\emptyset$$.

The maps $$f$$ and $$g$$ are in­verse to each other. In­deed, if we do $$f$$ and then $$g$$, we ob­tain a map from $$\emptyset$$ (be­ing the do­main of $$f$$) to $$\emptyset$$ (be­ing the image of $$g$$); but we know there’s a unique map $$\emptyset \to \emptyset$$, so we must have the com­po­si­tion $$g \circ f$$ be­ing equal to $$\mathrm{id}$$.

We’ve checked half of ”$$f$$ and $$g$$ are in­verse”; we still need to check that $$f \circ g$$ is equal to the iden­tity on $$X$$. This fol­lows by iden­ti­cal rea­son­ing: there is a unique map $$\mathrm{id}_X : X \to X$$ by the fact that $$X$$ satis­fies the uni­ver­sal prop­erty noteAnd we know that this map is the iden­tity, be­cause there’s always an iden­tity func­tion from any set $$Y$$ to it­self., but $$f \circ g$$ is a map from $$X$$ to $$X$$, so it must be $$\mathrm{id}_X$$.

So $$f$$ and $$g$$ are bi­jec­tions from $$\emptyset \to X$$ and $$X \to \emptyset$$ re­spec­tively.

# The mixture

This time, let us sup­pose $$X$$ is a set which satis­fies the uni­ver­sal prop­erty of the empty set. Then, in par­tic­u­lar, there is a (unique) map $$f: X \to \emptyset$$.

If we pick any el­e­ment $$x \in X$$, what is $$f(x)$$? It has to be a mem­ber of the empty set $$\emptyset$$, be­cause that’s the codomain of $$f$$. But there aren’t any mem­bers of the empty set!

So there is no such $$f$$ af­ter all, and so $$X$$ can’t ac­tu­ally satisfy the uni­ver­sal prop­erty af­ter all: we have found a set $$Y = \emptyset$$ for which there is no map (and hence cer­tainly no unique map) from $$X$$ to $$Y$$.

This method was a bit of a mix­ture of the two ways: it shows that a cer­tain map can’t ex­ist if we spec­ify a cer­tain ob­ject.

Parents:

• Universal property of the empty set

The empty set can be char­ac­ter­ised by how it in­ter­acts with other sets, rather than by any ex­plicit prop­erty of the empty set it­self.

• Universal property

A uni­ver­sal prop­erty is a way of defin­ing an ob­ject based purely on how it in­ter­acts with other ob­jects, rather than by any in­ter­nal prop­erty of the ob­ject it­self.