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 singleton set – sumset axiom singleton set   A set with exactly one element. singular cardinal   A cardinal that is not regular. slope   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 byThe 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. space   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. sphere   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) isThe 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. square   A regular polygon having four equal sides and four right angles. square matrix   A matrix that has the same number of rows as columns. stationary set   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. Stone-Weierstrass Theorem   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). story problem    ARTICLE   A mathematical problem presented as a real-world situation. See the article for problem solving techniques. subbase   In a topological space, a subbase is a collection of sets, the collection of all finite intersections of which constitutes a base. subfield   A subset of a field which is also a field, with the operations defined by restriction. subgroup   A subset of a group which is also a group. Every subgroup induces a partition of the original group, and if this partition is itself a group under the inherited group operation, the subgroup is called a normal subgroup. sublattice   We say that Y is a sublattice of a lattice X if Y is a subset of X and if a b, a b are in Y whenever a, b are in Y. subring   A subset of a ring which is also a ring, with the operations defined by restriction. subset   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.Cf. superset. Substitution Principle   Intuitively, the assertion that if one “plugs” a value into a formal polynomial, everything will work out as expected. Formally, the theorem that, given two commutative rings R and R', a homomorphism f from R to R', and an element r of R', there is a unique homomorphism F from the polynomial ring R[x] over R to R', with the properties F(a) = f(a) for all a in R,F(x) = r. For any polynomial p, Fr(p) is written p(r), and corresponds to “evaluating p on r.” subtract   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. successor   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.Cf. predecessor. sumset   Given a set A, the sumset of A, denoted byis the set containing all of the elements of the elements of A, that is, it is the union of the elements of A. sumset axiom   An axiom of set theory which states that if A is any set, then the sumset of A is also a set. singleton set – sumset axiom
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