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 join irreducible – logic 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 ifCf. join irreducible, meet prime, meet irreducible. 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, satisfyingx x = x = x x, andx (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. 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. 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. linear function   A polynomial function of degree one. The graph is a straight line. There are three special forms of the linear equation:In the first form, called the slope-interecept form, m is the slope of the line and b is the y-intercept. In the second form, called the point-slope form, m is the slope of the line and (x 0, y 0) is any point on the line. In the third form a is the x-intercept and b is the y-intercept. linear order   A set X is said to be a linear order if there is a relation “<” on its elements such that for all distinct x and y in X, either x < y or y < x, but not both. Sometimes also called a total order.Cf. partial order, dense linear order. linear space   See vector space. locus   A set of points satisfying a (geometric) condition. E.g., a parabola is the locus of points equidistant from a given point and a given line. log   See logarithm. logarithm   A logarithm is an expression of the form log bA, where b is called the base of the logarithm and A is called the argument, and it evaluates as the exponent to which the base must be raised to return the argument. Logarithms may be viewed as an alternative form of exponential notation, since we have log bA = x if and only if bx = A. A common logarithm is a logarith with base 10, and a natural logarithm (often written “ln”) is a logarithm with base e.Cf. laws of logarithms, Euler number. logarithmic function   A function which acts on its argument with a logarithm. logic   In the broadest sense, the term “logic” refers to the reasoning processes or forms of argument people use to reach valid conclusions from one or more premises (assumptions). More specifically, any particular logic is a system of inference, by which one reasons from premises to conclusions, or by which one validates one’s conclusions. An incorrect inference, i.e., one which violates the rules of logic, is called a fallacy, and conclusions reached by means of a fallacy are called invalid. (By contrast, when the premises of an argument are false but the argument is formally valid, then the conclusion is called unsound.)Ordinary logic falls into two broad types; deductive and inductive. Reasoning from general premises to a specific conclusion is deductive. For example, knowing that heavy clouds and a brisk breeze often signal an oncoming storm, one could conclude deductively that it might rain this afternoon. Reasoning from specific premises to a general conclusion, on the other hand, is inductive. For example, observing that every raven one sees is black, one might conclude inductively that all ravens are black. Deductive conclusions are necessarily true if the premises are true and the logic used formally valid (free of fallacy). Inductive conclusions are never certain, but are only more or less reliable. In mathematics only deductive reasoning is used (but see the entry for mathematical induction).Mathematicians use symbolic logic, that is, logic that can be reduced to a purely syntactical system which disregards the semantic content of any of the symbols in use. The two primary forms of symbolic logic are the propositional calculus and the predicate calculus. The latter augments the former with existential and universal quantifiers (permitting sentences that begin with “there exists ___ such that” and “for all ___”). When quantification is permitted only over elements of the universe of discourse, the logic is called “first-order.” When quantification over classes of objects and/or over predicates is permitted, it is called second-order logic.Cf. axiom, fuzzy logic, equational logic, syllogism, Hempel's Ravens Paradox. join irreducible – logic
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