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interior – line
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.
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.
isomorphism
A morphism that is both injective and surjective, that is, both “one-to-one” and “onto.” An isomorphism from a structure to itself is called an automorphism.
isotone function
A function that is order preserving and increasing.
Jordan Decomposition Theorem
If m is a signed measure, there exist unique positive measures m+ and m– such that m = m+ – m–, with m+ and m– mutually singular. This is called the Jordan decomposition of m, and the measures m+ and m– are called the positive and negative variations of m. The total variation of m is defined to be the sum of the positive and negative variations of m.
Latin
ARTICLE
Many Latin terms and phrases are used in mathematical writing. See the article for a full exposition.
laws of exponents
The following rules govern the behavior of exponents:
Additionally, x0 = 1 for all x except 0, and 00 = 0 or is left undefined (i.e., it is an indeterminate form).
Cf. rational exponent.
laws of logarithms
The following rules governing the behavior of logarithms are easily derived, and very useful in calculations:- log A + log B = log (A×B)
- log A – log B = log (A/B)
- log Ap = p × log A
Students often confuse these rules, so it is worth memorizing them as “the sum of the logs is the log of the product – and not the product of the logs,” etc.
least upper bound
An upper bound which is less than or equal to every other upper bound.
Cf. greatest lower bound.
least upper bound axiom
“Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.
Cf. supremum.
Lebesgue measure
The unique measure m on the real line, generated by the outer measure whose value on intervals is the length of the intervals, is called Lebesgue measure.
leg
On a right triangle, the sides adjacent to the right angle are called the legs.
Cf. hypotenuse.
limit
ARTICLE
The concept of a limit is central to our modern understanding of the differential and integral calculus. Naively, an expression involving a variable limits to some value L if it becomes arbitrarily close to L for appropriate choices of the variable. See the article for a full exposition.
limit point
Given a sequence of points ai , i = 1, 2, 3, ... , a point L is called a limit point of the sequence if every neighborhood of L contains all but finitely many of the ai .
If X is a metric space, then L is a limit point of X if a sequence from X may be chosen so that L is the limit point of that sequence. Limit points are not in general unique.
See also: accumulation point, perfect set.
line
Naively, “line” is a primitive concept, generally connoting a straight path through space. In mathematics this notion is abstracted formally in different ways depending on the type of mathematics under consideration.
Geometry: Euclid defined the term “line” as a point extended indefinitely in space. However, in modern Euclidean geometry, “line” is a primitive term, left purposely undefined, whose meaning is informed purely by the axioms in which the term appears.
Analytic geometry: Any set of ordered pairs (x, y) of all the points in the Cartesian plane satisfying an equation of the form ax + by + c = 0, where a, b, and c are real numbers. (There are three common and useful forms of this equation; see the entry for linear function.) This definition of the line may be generalized to a Euclidean space of n dimensions to include any set of ordered n-tuples (x1, x2, ..., xn) satisfying a1x1 + a2x2 + ... + anxn + c = 0. In polar coordinates a line is any set of all the ordered pairs (r, q) satisfying an equation of the form r = p sec (q - a), where p is the perpendicular distance from the pole to the line, and a is the angle of inclination of the line to the polar axis.
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