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 superset – transitive set superset   A set A is a superset of a set B if every element of B is an element of A.Cf. subset. supplemental angles   Two angles are supplemental if they add up to 180 degrees (p radians).Cf. complementary angles. supremum   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. surd   (rare) An irrational root of a number, e.g., the square root of two. Related article: Irrationality of the Square Root of 2 surjection   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. Suslin tree   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. symbolic logic   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. symmetric difference   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. symmetric relation   A relation “ ~ ” on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation “ ~ ” is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.Cf. reflexive relation, transitive relation. Tarski Truth Theorem   Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let Tj be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’)Tj. In other words, “truth” in a theory T is not definable in T. tesselation   A tiling of the plane, i.e. the use of plane figures to completely cover the plane without overlaps or gaps. A regular tesselation uses only a finite number of distinct shapes. Most regular tesselations are periodic, but some are aperiodic.Cf. polygon, Penrose tiles. Related MiniText: Mathematical Art of M.C. Escher tetrahedron   A polyhedron having four faces.The faces of a regular tetrahedron are congruent, equilateral triangles.Cf. Platonic solid. tiling   See tesselation. topology   Generally, topology is the study of those properties of a space which are invariant under continuous deformations, i.e., deformations which do not create “tears” or “holes.” More specifically, given a set X, a topology on X is a collection of subsets of X, called the open sets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersections and arbitrary (i.e., not necessarily finite or countable) unions. A set X with a topology defined upon it is called a topological space.Cf. homeomorphism. totally ordered set   A set with a total order defined on it. total order   An order relation “<” on a set S is a total order exactly if, for any two elements x and y of S, either x < y, x = y, or y < x, but no two of the three. Sometimes also called a linear order. Every totally ordered set is order-isomorphic to an ordinal, and this ordinal is called its order type (or sometimes just “type.”)Cf. partial order transcendental function   A function which is not an algebraic function, i.e., a function whose action on its argument(s) cannot be represented by the arithmetic and algebraic operations: addition and subtraction, multiplication and division, raising to a power, or extraction of roots. The exponential function, the logarithmic function, and the trigonometric functions are examples of transcendental functions. transcendental number   A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well. transitive closure   If x is any set, the transitive closure of x is the set containing all of the elements of x, and all of the elements of the elements of x, and all of the elements of the elements of the elements of x, and so on. Formally, the transitive closure Tc(x) of x is defined by the recursive formulaswhere w is the set of finite ordinals. transitive relation   A relation “ ~ ” on a set X is called transitive if it is the case that for every x, y, and z in X, if x ~ y and y ~ z , then x ~ z. For example, the relation “ < ” (less-than) on the set of natural numbers is transitive.Cf. reflexive relation, symmetric relation. transitive set   A set is called transitive if it is equal to its transitive closure. That is, x is a transitive set if whenever y is an element of x and z is an element of y, then z is an element of x. superset – transitive set
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