Tuples

Overview

Let $n\in \mathbb{N}$. An $n$-tuple is an ordered collection of $n$ elements, denoted $⟨x_1,\dots ,x_n⟩$. Two $n$-tuples are equal if and only if the have the same length and the same elements in the same order.

An $n$-tuple is alternatively called a list of length $n$. Sometimes the words tuple and list will be used without specifying a length. In both cases though, the tuple/list must have a finite, nonnegative length.

The $0$-tuple looks like $⟨⟩$.

Ordered Pairs

An ordered pair of $x$ and $y$, denoted $⟨x,y⟩$, is defined as

We define the first coordinate of $⟨x,y⟩$ to be $x$ and the second coordinate to be $y$. Similarly, ordered triples have form $⟨x,y,z⟩=⟨⟨x,y⟩,z⟩$, ordered quadruples have form $⟨x_1,x_2,x_3,x_4⟩=⟨⟨⟨x_1,x_2⟩,x_3⟩,x_4⟩$, and so on.

As a special case, we define the $1$-tuple $⟨x⟩=x$.