commutative convergent series
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.
A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative.
A ring in which the multiplication operation is commutative.
A test for the convergence of a series. See the article for a complete description.
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.
Two angles are complementary if they add up to a right angle.
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.
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.
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°.
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
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.
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.
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).
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.
An unvarying quantity, usually represented notationally by an alphabetic letter such as k, c, etc.
A constant function f is one whose value is the same at all points of its domain.
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.
In mathematics, the real numbers or real number line.
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.