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# Def/Finite sequence

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Definition of Finite sequence: Suppose that $n$ is a natural number, and $X$ is a set. A finite sequence of elements of $X$, of length $n$, is a function $a \colon n \rightarrow X$, where $n$ is thought of set-theoretically as: $$n = \{ 0,1,2,3, \ldots, n-1 \}.$$ If $0 \leq i < n$, then rather than writing $a(i)$, one uses the subscript notation $a_i$, to denote the evaluation of $a$ at $i$. One calls $a_i$ the $i^{th}$ entry in the finite sequence $a$.

When defining or declaring a finite sequence, one typically writes, "Let $a_0, \ldots, a_{n-1}$ be a sequence of..."

In some circumstances, it is useful to refer to the empty sequence of length $0$; such a sequence is unique, since there is a unique function from $\emptyset$ to a set $X$, namely $\emptyset$. For this reason, we write $\emptyset$ to denote the empty sequence, in this context.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Natural number, Def/Function

The following statements and definitions logically rely on the material of this page: Def/Circuit, Def/Domain topograph, Def/Euclidean algorithm, Def/Factorization into irreducibles, Def/Ordered tuple, Def/Prime factorization, State/Natural numbers can be factored into primes, State/Permutations can be decomposed into transpositions, State/Products of invertible residues are invertible, and State/Uniqueness of prime factorization

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Naive set theory, Clust/Sequences