Def/Finite sequence

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

Reversion

Suppose that $a_0, \ldots, a_{n-1}$ is a sequence of elements of $X$. The reversion, or reverse sequence is the sequence $a_{n-1}, \ldots, a_0$. More formally, the reverse sequence is the function $rev(a) \colon n \rightarrow X$ defined by: $$rev(a)_i = a_{n - (i + 1)}, \mbox{ for all } i \in n.$$

Entrywise Operation

Suppose that $a_0, \ldots, a_{n-1}$ is a sequence of elements of $X$. Suppose that $Y$ is another set, and $f \colon X \rightarrow Y$ is a function. Then one may apply $f$ entrywise to obtain a new sequence $f \circ a$, given by: $$(f \circ a)_i = f(a_i), \mbox{ for all } i \in n.$$

Similarly, if $a_0, \ldots, a_{n-1}$ is a sequence of elements of $X$, and $b_0, \ldots, b_{n-1}$ is a sequence of elements of $Y$, $Z$ is a set, and $g \colon X \times Y \rightarrow Z$ is a function, one may apply $g$ entrywise to the sequences $a$ and $b$ to obtain a sequence $g(a,b)$, given by: $$g(a,b)_i = g(a_i, b_i).$$

Concatenation

Suppose that $a_0, \ldots, a_{n-1}$ is a sequence of elements of $X$, of length $n$, and $b_0, \ldots, b_{m-1}$ is another sequence of elements of $X$ of length $m$. Then, one may concatenate the two sequences $a$ and $b$, to obtain a sequence $Conc(a,b)$ of length $n+m$, given by: $$Conc(a,b)_i = a_i \mbox{ if } 0 \leq i < n, \mbox{ and } Conc(a,b)_i = b_{i-n}, \mbox{ if } n \leq i < n+m.$$

Observe that $Conc(a, \emptyset) = Conc(\emptyset, a)$ for any sequence $a_0, \ldots, a_{n-1}$.