The Kuratowski ordered pair $(a,b):=\{\{a\},\{a,b\}\}$

For sets $a$ and $b$, the ordered pair $(a,b)$ is the set

$(a,b):=\{\{a\},\{a,b\}\},$

formed from the unordered pairs and singletons of The unordered pair $\{x,y\}$ and the singleton $\{x\}=\{x,x\}$; three applications of The Axiom of Pairing: $\mathrm{\forall }x\mathrm{\forall }y\mathrm{\exists }z\mathrm{\forall }t(t\in z\leftrightarrow (t=x\vee t=y))$ produce it. The first coordinate is $a$ and the second is $b$.

When $a=b$ the two members coincide, since $\{a,b\}=\{a,a\}=\{a\}$, and the pair degenerates to $(a,a)=\{\{a\}\}$.

• Why this set and not another. An ordered pair is required to satisfy one property, that $(a,b)=(c,d)$ exactly when $a=c$ and $b=d$; that is $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$, and it is the only thing any later construction uses. Other definitions with the same property exist, and nothing below distinguishes them from this one.

• The degenerate case is where a careless proof fails. An argument that treats $\{\{a\},\{a,b\}\}$ as a set with two distinct members breaks at $a=b$, and that case has to be handled separately in the proof of the characterising property.