inverse statement linear order
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.
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.
See reflexive relation.
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.
A function that is order preserving and increasing.
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.
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.
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.
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.
If a is an infinite cardinal and the cofinality of a is not greater than b, then ab is strictly greater than a.
Many Latin terms and phrases are used in mathematical writing. See the article for a full exposition.
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
A lattice is called complete if every subset of the lattice has a least upper bound and a greatest lower bound.
- x x = x = x x, and
- x (x y) = x = x (x y)
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:
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.
- log A + log B = log (A×B)
- log A – log B = log (A/B)
- log Ap = p × log A
least upper bound
An upper bound which is less than or equal to every other upper bound.
Cf. greatest lower bound.
On a right triangle, the sides adjacent to the right angle are called the legs.
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.
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.
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.
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.
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.