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.