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function – icosahedron
function
Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.
 More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively.
Fundamental Theorem of Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
Fundamental Theorem of Arithmetic
ARTICLE
Every natural number is either prime or may be decomposed uniquely into prime factors.
Fundamental Theorem of Calculus
Intitively, that the integral and derivative are inverse operatorations on functions. The fundamental theorem is commonly expressed in either one of two formal guises:
Cf. Riemann integral.
g.c.d.
Abbreviation for greatest common divisor.
genus
Topology: Given a topological surface S, the genus of S is the ordered pair (h, c) in which h is the number of handles on the surface and c is the number of cross-caps.
Graph Theory: The genus of a graph is the minimum genus of the topological surface on which the graph can be drawn without crossovers. Intuitively, the minumum number of handles one must attach to a sphere to draw the graph on it.
geometric series
An infinite series such that the ratio of consecutive terms is constant. Such a series may be denoted
where r is the common ratio. A geometric series converges if and only if the common ratio is less than 1, in which case the sum of the series is given by
Cf. arithmetic series, harmonic series.
Related article: Series
g.l.b.
Abbreviation for greatest lower bound.
graph
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Greek
ARTICLE
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
harmonic series
The infinite series whose terms are the reciprocals of the natural numbers
This series diverges, i.e., the sum does not exist.
Related article: Series
Hausdorff space
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heawood Map Coloring Theorem
For every positive integer n, the chromatic number of the orientable surface of genus n (sphere with n handles) is given by
For n = 0, this statement is exactly the Four Color Theorem, which was proven later, and does not bear Heawood's name.
Heine-Borel property
Every closed and bounded subset of the real numbers is compact.
homeomorphism
A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous.
hyperbola
The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Like the ellipse and parabola, the hyperbola is a conic section.
Related article: Conics
hypotenuse
On a right triangle, the side opposite the right angle.
i
See imaginary number.
icosahedron
A polyhedron having twenty faces.
The faces of a regular icosahedron are congruent, equilateral triangles.
Cf. Platonic solid.
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