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
|
|
interior – limit
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.
Kepler’s Laws
ARTICLE
The laws of planetary motion discovered by Johannes Kepler. See the article for an exposition.
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.
L’Hospital’s Rule
If a limit is in one of the indeterminate forms “infinity over infinity” or ”zero over zero,” then we have
That is, the limit is the same after taking the derivatives of both the numerator and denominator. L’Hospital’s Rule may be applied as many times as needed. Students often misapply the Rule by using it when the limit is not in one of the above indeterminate forms. This often results in an incorrect evaluation of the limit.
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.
|
|
 
|