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complete – converse statement
complete
Analysis: A metric space X is complete if every Cauchy sequence in X converges in X.
 Logic: a system of axioms for a mathematical theory is complete if every theorem in the theory is deducible from the axioms. Gödel's incompleteness theorem states that any axiom system which includes or allows the operations of arithmetic is necessarily incomplete.
Set Theory: If F is a filter on a set X and k is a regular, uncountable cardinal, then we say that F is k-complete (k-closed) if  A F for every A F with |A| < k. Every filter is w-complete. If k is the first uncountable cardinal ( 1), then F is called countably complete.
Related article: Gödel's Theorems
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 measure
See measure.
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.
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).
connected
A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty.
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...
continuum hypothesis
The claim that there is no set of intermediate cardinality between the set of natural numbers and the power set of the natural numbers (equivalently, between the set of natural numbers and the set of real numbers). This hypothesis was first formulated by Georg Cantor, who believed it to be true. It is now known to be independent of the usual axioms of set theory.
Cf. generalized continuum hypothesis.
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.
contrapositive
Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its contrapositive is “not B implies not A.” A conditional and its contrapositive are logically equivalent. Consequently, in mathematics, to prove a conditional it is a good strategy (and often easier) to prove its contrapositive instead.
Cf. inverse statment, converse statement.
convergent sequence
See sequence.
convergent series
See series.
Related article: Series
converse relation
If R is a relation, the relation R´ is called the converse relation of R if whenever xRy we have yR´x.
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.
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