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.
See algebra of sets.
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.
A function whose domain of definition is a collection of sets.
See ring of sets.
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.
A paradox of knowledge: is the distinction between matters of opinion and matters of fact a matter of opinion or a matter of fact? See the article for discussion.
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 set with exactly one element.
A cardinal that is not regular.
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 matrix that has the same number of rows as columns.
If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a.
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.
Given a set A, the sumset of A, denoted by
is the set containing all of the elements of the elements of A, that is, it is the union of the elements of A.