BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry		graph – identity map 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. group   A set together with a binary operation defined on its elements, satisfying 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. 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. Cf. field, module, ring group automorphism   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. group homomorphism   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. 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. groupoid   See group. 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 Hempel’s Ravens Paradox    ARTICLE   A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition. homomorphism   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. 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 hyperset   A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}. 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. idempotent   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. identity element   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. identity function   A function that maps each domain element to itself. Also called the identity map. identity map   See identity function.
		graph – identity map

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