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		hyperbola – injective 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. image   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. imaginary number   By definition, the square root of –1, i.e., i 2 = –1. Cf. complex number. inaccessible cardinal   A cardinal k is called weakly inaccessible if and only if it is a regular limit cardinal. A cardinal k is called strongly inaccessible if and only if k > w and for every cardinal l less than k, 2l is less than k. If GCH is assumed, then weakly inaccessibles are strongly inaccessible. Ordinary set theory (ZFC) cannot prove that weakly inaccessibles exist. Cf. Kφnig’s Lemma. incidence matrix   A way to represent a graph as a matrix. Given a graph G with vertices {v1, v2, ..., vn} and edges {e1, e2, ..., em}, the incidence matrix of G is an n by m matrix, whose i, j entry is the number of times ej is incident on vi (zero if not incident, one if incident and not a loop, two if incident and a loop). Cf. adjacency matrix. independent axiom   In a formal mathematical theory, a formula or statement in the theory is said to be independent if it cannot be derived (proved, deduced) from the other axioms of the theory. indeterminate form   A limit of an expression is said to be indeterminate, or in indeterminate form, if when evaluated directly it resolves to one of the forms Such limits may often be evaluated by manipulating them algebraically before applying the limit, or, in the case of the first two indeterminate forms shown, by applying L'Hospital's Rule. Related article: Limits inf   Abbreviation of infimum. inferential statistics   Those statistical methods that are used to make inferences about a population from a sample chosen to represent that population. Cf. descriptive statistics. infimum   The infimum of any subset of a linear order (linearly ordered set) is the greatest lower bound of the subset. In particular, the infimum of any set of numbers is the largest number in the set which is less than or equal to every other number in the set. In a complete linear order the infimum of any bounded set always exists. Cf. supremum, least upper bound axiom. infinity   Infinity is a concept understood in different ways depending upon the context in which the word is used. In particular, infinity is not a number in the ordinary sense. The so-called extended real numbers include either a positive or a negative infinity (but not both). When this is done, the algebraic forms "infinity plus negative infinity", "infinity times zero", and "infinity divided by infinity" are undefined. Infinite ordinals may be countable or uncountable. Whether actually infinite totalities may be admitted, used, or analyzed remains a contentious issue in the philosophy of mathematics. See the minitext for a thorough treatment. Related MiniText: Infinity -- You Can't Get There From Here... injection   An injective function, i.e., a function that is “one-to-one.” Equivalently, a function that maps exactly one element of its domain to each element of its range. Cf. surjection, bijection. injective   A function is injective, also called “one-to-one,” if to each element of the range at most one element of the domain is mapped by the function. Cf. surjective, bijective.
		hyperbola – injective

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