# Definitions > [!NOTE] Definition 1 (List of given Length) > Let $A$ be a [[Sets|set]]. Suppose $n\in \mathbb{N}$ is a [[Natural Numbers|natural number]]. A *list $L$ of length $n$ (or $n$-tuple or finite sequence) over (or whose elements are taken from) $A$* is a [[Function|function]] $L:\{ k\in\mathbb{N} \mid k <n \} \to A$ > **Notation**: Suppose that $C$ is a list of length $n$ taken from the set $A.$ Then we may write either $C=(a_{k})_{k=0}^{n-1}$ or $C=(a_{0},a_{1},a_{2},\dots,a_{n-1}).$ The same notation is used for [[Ordered pair|ordered pair]] thus we may define a list as below: > [!NOTE] Definition 2 (List of given length) > Suppose $A$ is a set. Let $n\in \mathbb{N}.$ A list of length $n$ whose elements are taken from $A$ is an element of $A^{k},$ which the denotes the [[Cartesian Product|cartesian product]] $\underbrace{A \times A \times\dots \times A}_{k \text{ times}}.$ # Properties By [[Number of Lists of Length k whose Elements are Taken From a Finite Set]], $|A^{k}|=|A|^{k}$ where $A$ is finite. **k-Permutation**: A list of length $k$ whose elements are taken from $A$ without repeats is a [[Partial Permutation of n Letters (Ordered Selection)|k-permutation]] of $A.$ **Strings**: A list taken from a finite set is known as a [[String|string]]. In this case, the codomain of the list is called an alphabet. **List of reals**: [[Erdös-Szekeres Theorem|Erdös-Szekeres Theorem]] gives a lower bound for the length of the longest monotonic sublist of a sublist. A list of reals is a [[Euclidean Vector|Euclidean vector]]. # Applications **Counting**: Suppose $S$ is a set. If there is a surjective list of some length whose elements are taken from $S$ then $S$ is [[Finite Set|finite]]. In this case, the [[Cardinality|cardinality]] of the set is the length of the list and is [[Uniqueness of cardinality of finite sets|unique]]. **Generalisations**: Informally, a [[Sequences|sequence]] is an infinite list.