BROWSE
ALPHABETICALLY
LEVEL:
Elementary
Advanced
Both
INCLUDE TOPICS:
Basic Math
Algebra
Analysis
Biography
Calculus
Comp Sci
Discrete
Economics
Foundations
Geometry
Graph Thry
History
Number Thry
Phys Sci
Statistics
Topology
Trigonometry
|
|
commutative – convergent series
commutative
An operation · on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.
 Cf. associative, distributive property.
commutative property
A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative.
commutative ring
A ring in which the multiplication operation is commutative.
comparison test
ARTICLE
A test for the convergence of a series. See the article for a complete description.
complement
The complement of a set A is the set of all elements that are not elements of A.
Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G.
complementary angles
Two angles are complementary if they add up to a right angle.
complete lattice
A lattice X is complete if every subset of X has a least upper bound and a greatest lower bound in X.
complete ordered field
A linearly ordered field in which every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers.
complex number
An element of the set C of numbers of the form a + bi, where a and b are real numbers and i 2 = -1. The number i is called the imaginary number.
concave
A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.
Cf. convex.
concave function
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
conditional statement
A statement of the form if A then B, or A implies B. A conditional is equivalent to its contrapositive, not B implies not A. See also: inverse statement and converse statement.
cone
ARTICLE
The infinite surface of revolution generated as shown
The term also refers to the solid bounded by one of the nappes and a flat elliptical base. If in this case the base is circular (at right angles to the axis), the cone is called a right circular cone. The surface area S (excluding the base) and volume V of a right circular cone are given by
Cf. conic section.
congruence relation
Given a structure X (i.e., X is a space, algebra, etc.), then an equivalence relation ~ on X is called a congruence relation if it is preserved by every defined operation on X. That is, for any x, y in X, if x ~ y and f is defined on X, then f(x) ~ f(y).
conic section
ARTICLE
A plane curve, either the ellipse, parabola, or hyperbola, which results from the intersection of a plane with a cone. See the article for a full exposition
Cf. Dandelin's Spheres.
constant
An unvarying quantity, usually represented notationally by an alphabetic letter such as k, c, etc.
constant function
A constant function f is one whose value is the same at all points of its domain.
continuous
Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).
Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.
Cf. uniformly continuous, equicontinuous, absolutely continuous.
continuum
In mathematics, the real numbers or real number line.
Related article: Gödel's Theorems
Related MiniText: Infinity -- You Can't Get There From Here...
contradiction
An assertion that some statement is simultaneously true and false.
When a contradiction arises in mathematics, it is an indication that a mistaken assumption has been made, and this is often used to write proofs by contradiction, a strategy in which the negation of the statement to be proved is assumed, and a contradiction then derived. The contradiction is then taken as proving the original (non-negated) statement. (This strategy relies on the supposition that the mathematical theory under discussion is itself consistent, that is, free of contradiction.)
It sometimes happens that a contradiction arises, and yet none of the assumptions leading to the contradiction can be sacrificed. This is known as a paradox. If the assumptions are formal axioms, the result is called an antinomy.
convergent series
See series.
Related article: Series
|
|
 
|