For any $a$ and $b$ there exists a set $\{ a, b \}$ that contains exactly $a$ and $b$ .

The Axiom of Pairing is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:$$ \forall a \forall b \exists c \forall x (x \in c \leftrightarrow x = a \lor x = b).$$ Using the Axiom of Extensionality, we see that the set $c$ is unique, so it makes sense to define the pair$$ \{ a, b \} = \mbox{ the unique } c \mbox{ such that } \forall x (x \in c \leftrightarrow x = a \lor x = b).$$

Using the Axiom of Pairing, we may define, for any set $a$ , the singleton$$ \{ a \} = \{ a, a \}.$$

We may also define, for any set $a$ and $b$ , the ordered pair$$ (a, b) = \{ \{ a \}, \{ a, b \} \}.$$

Note that this definition satisfies the condition$$ (a, b) = (c, d) \mbox{ iff } a = c \mbox{ and } b = d.$$

We may define the ordered $n$ -tuple recursively$$ (a_1, \ldots, a_n) = ((a_1, \ldots, a_{n-1}), a_n).$$