sphere symmetric difference
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.
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.
If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X).
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
In a topological space, a subbase is a collection of sets, the collection of all finite intersections of which constitutes a base.
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.
An axiom of set theory which states that if A is any set, then the sumset of A is also a set.
Abbreviation of supremum.
A set A is a superset of a set B if every element of B is an element of A.
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
(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.
A function f from a set X to a set Y is surjective, also called “onto,” if to each element y of Y there is an element x of X such that f maps x to y, i.e., f (x)=y. Compare: injective, bijective.
Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that |T| = a, and every chain and every antichain of T has cardinality less than a.
Cf. Aronszajn tree.
Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus.
The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.
Cf. intersection, union.