Tuple

Definition 1. The tuple of sets $x$ and $y$ is

\[(x, y) = \{\{x\}, \{x, y\}\}.\]

For sets $a$, $b$, $c$, $d$, etc., we define

\[\begin{align*} (a, b, c) &= ((a, b), c), \\ (a, b, c, d) &= ((a, b, c), d), \\ &\text{etc.} \end{align*}\]

An $n$-tuple is a tuple of $n$ sets.

Proposition 1 (Orderedness). For any sets $a_1$, $\cdots$, $a_n$ and $b_1$, $\cdots$, $b_n$ ($n\geq 2$),

\[(a_1, \cdots, a_n) = (b_1, \cdots, b_n) \enspace\lrimp\enspace% a_1 = b_1 \land \cdots \land a_n = b_n.\]