right angle surjection
An angle of 90 degrees (p/2 radians). Equivalently, it can be said that two right angles are supplemental angles, i.e., they add up to a straight line (180 degrees or p radians).
Cf. complementary angles, acute, obtuse.
An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.
Of a polynomial p: A number x such that p(x) = 0.
A quantity having only magnitude, not direction (typically an element of a field, such as the real numbers or complex numbers).
A triangle is called scalene if all of its sides are unequal (equivalently, if all of its angles are unequal).
A number is written in scientific notation when it is written as the product of a real number between 1 and 10 and a power of 10. E.g., 320 is written in scientific notation as 3.2 × 102. On some calculators and in some textbooks, this may be written as 3.2E2. Scientific notation is a convenient way to represent very large and very small numbers.
A sequence is a set (of numbers, or sets, or functions, etc.) indexed by the natural numbers. Sequences may be infinite, and may be regarded as a function with domain the set of natural numbers and range the set of objects in the sequence.
An infinite sequence of numbers is said to converge to a number L provided that, given any positive e, we may find a natural number N such that for all terms of the sequence after the N th one, their difference from L is less than e. Naively, the terms of the sequence eventually become “arbitrarily close” to L. Such a sequence is called convergent, and the number L is called the limit of the sequence, or the limit point, or sometimes the accumulation point of the sequence.
Alternatively, a cluster point or accumulation point P of a sequence may be defined as a point with the property that infinitely many terms of the sequence lie in any neighborhood of P. A sequence may have more than one such cluster point (even infinitely many).
A sequence is called Cauchy if, for every e greater than zero, we may find a natural number N so that the difference between any two terms following the N th term is smaller than e. Every convergent sequence is Cauchy; the converse is true in complete spaces.
Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.
Cf. ZF, ZFC.
The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B.
Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.
Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).
Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc.
Graph Theory: Two vertices or edges of a graph are called similar if there is an automorphism of the graph that takes one to the other.
A line in the Cartesian plane which passes through two points (x 1, y 1) and (x 2, y 2) has a slope m given by
The slope may easily be remembered as “rise over run.” It is evident that the slope of a horizontal line is 0, and the slope of a vertical line is undefined.
Cf. linear function.
Any abstract set with a structure defined on it, such as an order relation, metric, etc.
Cf. Euclidean space, Hilbert space, metric space, topological space.
A closed surface, all points of which are equidistant from a given point, called the center.
In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) is
The term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.
More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1.
A regular polygon having four equal sides and four right angles.
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
(rare) An irrational root of a number, e.g., the square root of two.
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.