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sequence – surd
sequence
A sequence is a set (of numbers, or sets, or functions, etc.) indexed by the natural numbers. Sequences may be infinite, and may be regarded as a function with domain the set of natural numbers and range the set of objects in the sequence.
 An infinite sequence of numbers is said to converge to a number L provided that, given any positive e, we may find a natural number N such that for all terms of the sequence after the N th one, their difference from L is less than e. Naively, the terms of the sequence eventually become “arbitrarily close” to L. Such a sequence is called convergent, and the number L is called the limit of the sequence, or the limit point, or sometimes the accumulation point of the sequence.
 Alternatively, a cluster point or accumulation point P of a sequence may be defined as a point with the property that infinitely many terms of the sequence lie in any neighborhood of P. A sequence may have more than one such cluster point (even infinitely many).
A sequence is called Cauchy if, for every e greater than zero, we may find a natural number N so that the difference between any two terms following the N th term is smaller than e. Every convergent sequence is Cauchy; the converse is true in complete spaces.
Cf. series.
Related article: Limits
series
ARTICLE
A series is an infinite sum, where the nth summand is the nth term of a sequence. A series is usually denoted using “sigma notation,” i.e.,
The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience. The nth partial sum Sn of a series is the (finite) sum of the first n terms of the series. A series is said to converge if and only if its sequence of partial sums {S 1, S 2, . . . , Sn, . . . } is a convergent sequence. There are several important types of series and several tests for the convergence of a series. Additionally, most useful functions have Taylor series representations, which makes them very important in the study of differential equations. See the article for a complete description.
set
Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.
Cf. ZF, ZFC.
set difference
The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B.
set theory
Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.
Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).
Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc.
signed measure
Given a set X together with a s-algebra of sets M defined on it, a signed measure on (X, M) is an extended real-valued function m with domain M satisfying: - The signed measure of the empty set is zero.
- The signed measure m assumes at most one of the values +/- infinity.
- (Countable additivity) Given a countable sequence of disjoint sets in M, the signed measure of the union of the sequence is equal to the sum of the signed measures of the sets in the sequence, where this sum converges absolutely if the signed measure of the union is finite.
Technically speaking, every measure is a signed measure; ordinary (i.e., nowhere negative) measures are sometimes called positive measures.
similar
Graph Theory: Two vertices or edges of a graph are called similar if there is an automorphism of the graph that takes one to the other.
slope
A line in the Cartesian plane which passes through two points (x 1, y 1) and (x 2, y 2) has a slope m given by
The slope may easily be remembered as “rise over run.” It is evident that the slope of a horizontal line is 0, and the slope of a vertical line is undefined.
Cf. linear function.
space
Any abstract set with a structure defined on it, such as an order relation, metric, etc.
Cf. Euclidean space, Hilbert space, metric space, topological space.
sphere
A closed surface, all points of which are equidistant from a given point, called the center.
In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) is
The term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.
More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1.
square
A regular polygon having four equal sides and four right angles.
square matrix
A matrix that has the same number of rows as columns.
Stone-Weierstrass Theorem
If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X).
story problem
ARTICLE
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
subset
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
Cf. superset.
subtract
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
successor
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Cf. predecessor.
sup
Abbreviation of supremum.
supplemental angles
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
supremum
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
surd
(rare) An irrational root of a number, e.g., the square root of two.
Related article: Irrationality of the Square Root of 2
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