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commutative – cycle
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.
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.
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.
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.
converse statement
Given a conditional, i.e., a statement of the form if A then B, or A implies B, its converse is B implies A. A conditional neither implies nor is implied by its converse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other.
convex
Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)
More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor convex.
Cf. concave.
cross product
See vector product.
cube
A regular polyhedron having six square faces.
More generally, an n-dimensional cube in the first quadrant of a Euclidean space with one vertex at the origin is given by the collection of all n-tuples of the form (e1,e2, ... , en), with each ei an element of the set {0,1}.
Cf. platonic solid.
Related article: Platonic Solids
cycle
Algebra: See permutation.
Graph Theory: A graph C whose vertices and edges can be numbered {v1, v2, ..., vn} and {e1, e2, ..., en}, respectively, for some n  1, such that ei is incident on vi and vi + 1 for every i < n, and en is incident on vn and v1. A cycle with n vertices is denoted by Cn. For a given graph G, a cycle of G (or a cycle in G) is a subgraph of G that is also a cycle. Cycles are sometimes also called circuits.
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