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indeterminate form – irreflexive
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.
integer
An element of the set Z consisting of the natural numbers, zero, and the additive inverses (negatives) of the natural numbers. I.e., Z = { ... -3, -2, -1, 0, 1, 2, 3, ... }. The use of Z to denote the set of integers stems from the German word zahlen, which means “to count.”
Cf. natural number.
Related MiniText: Number -- What Is How Many?
integral
An antiderivative of a function. That is, if f(x) is a real-valued function, an antiderivative F(x) of f(x) has the property that the derivative of F with respect to x is f.
The definite integral (called the Riemann integral) of a real-valued function f(x) from x = a to x = b is the limit of the Riemann sum:
assuming this limit exists, where ci is in the i th subinterval of the partition of (a, b), and where a is the lower limit, and b is the upper limit, of the integral.
Cf. fundamental theorem of calculus, Lebesgue integral.
integral test
A test for the convergence of a series. See the related article for a complete description.
Related article: Series
integration
Obtaining an integral of a function.
integration formulas
ARTICLE
See the article for a complete list of common integration formulas.
Cf. integral.
interior
The interior of a subset E of a topological space is the largest open set contained in E. It may also be expressed as the intersection of E with the closure of its complement. If E is open, then it is equal to its interior. If E contains no open sets we say that it has empty interior, and this equivalent to saying that E is nowhere dense.
intersection
The intersection of two sets A and B is the set of those elements common to both A and B, and is denoted by
Thus, an element x belongs to the intersection only if it is an element of A and also an element of B. If the intersection is taken over a family of sets {Ai}i = 1, 2, ... n, then it is the set of those elements that are in every set in the family, denoted
Sometimes one speaks of the interesection over a (single) set A, and this indicates the set of elements that are in every element of A:
When considering an algebra of sets, the intersection of two or more sets is sometimes called the meet of the sets.
Cf. union.
interval
The set of all real numbers lying between two given real numbers a and b. If both a and b are included in the set it is called a closed interval. If neither a nor b is included it is called an open interval. If only one of a or b is included the interval is called half-open (equivalently, half-closed). Intervals may be denoted using either interval notation or set notation, as follows:
Although intervals are most commonly intervals on the real line, the definition carries over without modification to any totally ordered set.
interval graph
An intersection graph of a finite family of intervals on the real line.
inverse function
If f is an injective (i.e., one-to-one) function, then f is said to be invertible, and its inverse is the function g satisfying g(f(x)) = f(g(x)) = x for every x in the domain of f.
inverse image
Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y.
inverse statement
Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its inverse is “not A implies not B.” A conditional neither implies nor is implied by its inverse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other.
irrational number
A real number which is not a rational number, i.e., which cannot be expressed as a ratio of integers. Examples: p (the ratio of the circumference of a circle to its diameter) and the square root of 2 (ratio of the length of the diagonal of a square to the length of one side). The irrational numbers are uncountably infinite.
Related article: Irrationality of the Square Root of 2
Related MiniText: Number -- What Is How Many?
irreflexive
See reflexive relation.
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