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conditional statement – dense
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).
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...
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.
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.
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.
convex set
A subset X of a partially ordered set is convex if for any elements a, b in X such that a  b, then all elements x satisfying a x b are also in X.
countable
A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers.
Related MiniText: Infinity -- You Can't Get There From Here...
countable chain condition
See chain condition.
countably infinite
See countable
cover
Topology: a collection of sets which contains a given set. If the sets in the covering collection are open sets, the cover is called an open cover.
Partially ordered sets: If x and y are elements of a partially ordered set such that x  y, and such that there is no z such that x z y, then we say that x covers y.
Graph Theory: An edge or vertex of a graph is said to cover (verb) those vertices or edges, respectively, that it is incident on. A set of edges or vertices is said to cover any vertex or edge covered by any element of that set. A set of edges or vertices that covers all the vertices or edges, respectively, of the graph is called a cover (noun), usually with a specification of whether it consists of edges or vertices. See edge cover, vertex cover.
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
cub set
Closed, unbounded set. See closed set.
D-system
Set Theory: A family A of sets is called a D-system (also called a quasi-disjoint family) if and only if there is a fixed set r, called the root of the D-system, such that for any two elements x and y of A, x  y = r. It is a theorem that every uncountable family of finite sets has a sub-family which forms a D-system.
Cf. almost disjoint.
Dedekind cut
A partition of the set of rational numbers into two subsets A and B, such that every element of A is less than every element of B. The set of real numbers can be formally defined as the set of all Dedekind cuts; e.g., we can take the real number “square root of 2” to be the cut
Arithmetic on real numbers is then defined in terms of operations on cuts. For example, if (A1, B1) and (A2, B2) are Dedekind cuts defining real numbers x and y, then we define the inequality x < y to mean that there is an element of A2 that is not an element of A1, and we define x + y to be the real number corresponding to the Dedekind cut (A, B) where the elements of A are all those rational numbers obtained by adding an element of A1 to an element of A2.
The set of irrational numbers can then be defined as the set of all real numbers that cannot be represented as a ratio of integers.
degree
Geometry: A unit of measure of angles, denoted by “ ° ”. A complete circle contains 360°, a straight line 180°, and a right angle 90°. Compare radians.
Algebra: The degree of a polynomial is the highest sum of exponents appearing in any term with a non-zero coefficient.
Graph Theory: The degree of a vertex is the number of edges incident on that vertex, counting loops twice. The degree of an edge is an unordered pair of the degrees of its two ends.
dense
Given a space X and a subset A of X, we say that A is dense in X if the intersection of every open set of X with A is non-empty. E.g., the rational numbers are dense in the real numbers.
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