graph imaginary number
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.
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
A set together with a binary operation defined on its elements, satisfying
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
- The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a΄, called the inverse of a, so that aa΄ = a΄a = e.
Cf. field, module, ring
A group isomorphism from a group to itself. That is, a bijective function from a group to itself that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group isomorphism.
A function from one group to another that preserves the group operation, i.e., f(ab) = f(a)f(b) for all a, b.
Cf. group automorphism, group isomorphism.
A group homomorphism that is both one-to-one and onto. That is, a bijective function from one group to another that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group automorphism.
The infinite series whose terms are the reciprocals of the natural numbers
This series diverges, i.e., the sum does not exist.
A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).
Cf. group homomorphism, ring homomorphism.
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.
On a right triangle, the side opposite the right angle.
See imaginary number.
A polyhedron having twenty faces.
The faces of a regular icosahedron are congruent, equilateral triangles.
Cf. Platonic solid.
An element of an algebra is called idempotent if it is equal to its own square.
If for a binary operation · on an algebra A we have that a · a = a for every a in A, then · is called an idempotent operation.
Given an algebra X with a binary operation , an identity element of X is an element e such that for all x in X, x e = e x = x. If is addition, the identity element is usually called zero and denoted by 0; if the operation is multiplication, the identity element is usually called unity and denoted by 1.
It is possible for an algebra to have distinct left and right identities, that is, distinct e and e such that for all x, e x = x and x e = x. If an element is both a left and right identity, then it is the unique identity element in the algebra for that operation.
A function that maps each domain element to itself. Also called the identity map.
See identity function.
Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f.
By definition, the square root of 1, i.e., i 2 = 1.
Cf. complex number.