Definition:Sequence
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Informal Definition
A sequence is a set of objects which is listed in a specific order, one after another.
Thus one can identify the elements of a sequence as being the first, the second, the third, ... the $n$th, and so on.
Formal Definition
A sequence is a mapping whose domain is a subset of $\N$.
It can be seen that a sequence is an instance of a family of elements indexed by $\N$.
Terms
The elements of a sequence are known as its terms.
Finite Sequence
A finite sequence is a sequence whose domain is finite.
Length of a Sequence
The length of a finite sequence is the number of terms it contains, or equivalently, the cardinality of its domain.
Sequence of n Terms
A sequence of $n$ terms is a sequence whose domain has $n$ elements.
Such a sequence is also known as an ordered n-tuple.
Null Sequence
A null sequence (or empty sequence) is one containing no terms.
Thus it is a mapping from $\varnothing$ to $S$ and therefore is null.
Infinite Sequence
An infinite sequence is a sequence whose domain is infinite.
Codomain
The codomain of a sequence can be elements of a set of any objects.
If the codomain of a sequence $f$ is $S$, then the sequence is said to be a sequence of elements of $S$, or a sequence in $S$.
Rational Sequence
A rational sequence is a (usually) infinite sequence whose codomain is the set of rational numbers $\Q$.
Real Sequence
A real sequence is a (usually) infinite sequence whose codomain is the set of real numbers $\R$.
Notation
The notation for a sequence is as follows.
If $f: A \to S$ is a sequence, then a symbol, for example "$a$", is chosen to represent elements of this sequence.
Then for each $k \in A$, $f \left({k}\right)$ is denoted $a_k$, and $f$ itself is denoted $\left \langle {a_k} \right \rangle_{k \in A}$.
Other types of brackets may be encountered, eg. $\left({a_k}\right)_{k \in A}$ and $\left\{{a_k}\right\}_{k \in A}$.
The latter is discouraged because of the implication that the order of the terms does not matter.
Any expression can be used to denote the domain of $f$ in place of $k \in A$.
The set $A$ is usually understood to be the set $\left\{{1, 2, 3, \ldots, n}\right\}$.
If this is the case, then it is usual to write $\left \langle {a_k} \right \rangle_{k \in A}$ as $\left \langle {a_k} \right \rangle$ or even as $\left \langle {a} \right \rangle$ if brevity and simplicity improve clarity.
Sequence of Distinct Terms
A sequence of distinct terms of $S$ is an injection from a subset of $\N$ into $S$.
A sequence $\left \langle {a_k} \right \rangle_{k \in A}$ is a sequence of distinct terms iff $a_j \ne a_k$ for all $j, k \in A$ such that $j \ne k$.
Equality of Sequences
Let $f$ and $g$ be two sequences:
- $f = \left({x_1, x_2, \ldots, x_n}\right)$
- $g = \left({y_1, y_2, \ldots, y_m}\right)$
Then $f = g$ iff:
- $m = n$
- $\forall i: 1 \le i \ne n: x_i = y_i$
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers
- Seth Warner: Modern Algebra (1965): $\S 18$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$: Example $13$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): $\S 1.2$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.2$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.10, \ \text{A}.12$
