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irrational number – limit point
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.
join
A binary operation whose value on two elements a and b of a lattice is the least upper bound of a and b, denoted a  b.
Cf. meet.
join irreducible
An element a of a lattice is called join irreducible if whenever a = b c then a = b or a = c. More generally, a is called strictly join irreducible if whenever a is the join of a subset X of the lattice, then a is an element of X.
Cf. join prime, meet irreducible, meet prime.
join-morphism
A function f on a lattice L is called a join-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a join-morphism, then f is called a join-isomorphism. A join-isomorphism from a lattice to itself is called a join-automorphism.
Cf. join, meet-morphism.
join prime
An element a of a lattice is called join prime if whenever a  b c then a b or a c. Also, a is called strictly join prime if
Cf. join irreducible, meet prime, meet irreducible.
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.
König’s Lemma
If a is an infinite cardinal and the cofinality of a is not greater than b, then ab is strictly greater than a.
Latin
ARTICLE
Many Latin terms and phrases are used in mathematical writing. See the article for a full exposition.
lattice
A partially ordered set in which each pair of elements has a least upper bound, called the join, and a greatest lower bound, called the meet. The join of two elements x and y is denoted by x y, and the meet by x  y.
A lattice may be equivalently defined as a set with an algebra having two binary operations (meet) and (join) defined on it, satisfying- x x = x = x x, and
- x (x y) = x = x (x y)
A lattice is called complete if every subset of the lattice has a least upper bound and a greatest lower bound.
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.
liar paradox
According to a legend found in many ancient sources (including a letter of St. Paul), there was a certain Cretan named Epimenides who proclaimed, “All Cretans are liars.” This statement involves a contradiction on its face, since if all Cretans are liars and Epimenides is a liar, then the statement must be false, that is, not all Cretans are liars.
There are many versions of the liar paradox, and many puzzles based upon it. The simplest form is, “This statement is false,” a statement that’s true if it’s false and false if it’s true.
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.
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