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 group – identity function 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 numbersThis 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. Hempel’s Ravens Paradox    ARTICLE   A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition. hexagon   A regular polygon having six equal sides and six equal interior angles. Hexagons can be used to tile the plane, and do so more efficiently than any other polygon.Cf. tesselation. Hilbert’s Problems    ARTICLE   23 problems posed by Hilbert to the community of mathematicians at the Paris conference of 1900. See the article for details. homeomorphism   A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous. 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. 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. group – identity function
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