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Cartesian plane – complementary angles
Cartesian plane
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
Cartesian product
For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian product
is the set of ordered n-tuples (a1,a2, ... ,an) with a1 an element of A1, a2 an element of A2, etc. The Cartesian product R2 of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rn. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice.
Cauchy sequence
A sequence (x1, x2, x3, ... ) of elements of a metric space X with metric d(x, y) is Cauchy if for any e greater than zero there is some natural number N such that
In other words, in a Cauchy sequence, the elements eventually become “arbitrarily close together.” If the metric space X is closed, this condition is equivalent to the sequence being convergent.
characteristic function
Given a subset E of a space X, the characteristic function cE is defined by cE(x) = 1 if x is in E, and cE(x) = 0 otherwise. All properties of sets and set operations may be expressed by means of characteristic functions.
choice, axiom of
See: axiom of choice.
circle
In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.
The interior of a circle is referred to as an open disk.
A circle is also a conic section; a special case of an ellipse in which the foci coincide.
Related article: Conics
circumference
Geometry: The distance around a circle in the plane, or around a great circle of a sphere.
Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles.
closed
General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.
Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.
See the following listings for other uses of the word “closed” in mathematics.
Topology: A set is topologically closed if it is not open.
closed interval
An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. open interval.
closed set
Topology: A subset E of a topological space X is closed if X - E (set difference) is open. In a metric space, E is closed if every convergent sequence in E converges in E; equivalently, if every accumulation point of E is in E.
Set Theory: If a is a limit ordinal, then a set C contained in a is called closed if and only if for every limit ordinal b less than a, if C  b is unbounded in b, then b  C. C is called c.u.b. (“cub set” or “club set”) if and only if C is closed and unbounded in a.
Cf. stationary set.
closure
Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.
Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G.
closure operator
If X is a set, then a function C from P(X) into P(X) (i.e., a function on the power set of X) is called a closure operator provided- Y is contained in C(Y) for every subset Y of X,
- C(C(Y)) = C(Y) for every subset Y of X, and
- If Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z).
Closure operators induce closed set systems.
coefficient
See polynomial.
combination
A subselection of a set of r elements from a set of n elements. The number of such combinations, i.e., the number of ways in which r elements may be chosen from a set of n elements, is given by the formula
This operation is also sometimes denoted by nC r, and is read “n choose r.”
Cf. permutation, factorial.
common logarithm
A logarithm with base 10.
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.
compact
Topology: In a topological space, a set E is compact if every open covering of E has a finite subcover, i.e., a finite subcollection which also covers E. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. E is called s-compact if there exists a sequence of compact sets {Ci} such that E is contained in their union.
Cf. locally compact, Bolzano-Weierstrass property, Heine-Borel property.
Set Theory: A cardinal k is called weakly compact if it is uncountable and
Equivalently, k is weakly compact if it is strongly inaccessible and there are no k-Aronszajn trees.
Lattices: an element a of a lattice L is called compact if whenever a is dominated by the join of a subset X of L then a is dominated by the join of a finite subset of X. Symbolically:
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.
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