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total order – unordered pair
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
total variation
(For total variation of a signed measure, see the Jordan Decomposition Theorem.) If f is a function on the real numbers with range in the complex numbers, then the function Tf given by
is called the total variation function of f, where here the supremum is being taken over all finite partitions of the real line up to x. If the limit of this function as x goes to infinity is finite, then f is said to be of bounded variation. The space of all such functions is usually denoted by BV. Functions which are increasing and bounded are in BV, and differentiable functions whose derivative is bounded are in BV.
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.
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 formulas
where 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.
tree
Graph Theory: A connected graph that does not contain any cycles. So named because they vaguely resemble trees in appearance.
Cf. forest.
Set Theory: A partial order T in which, for each x in T, the set {y T: y < x} is well-ordered. The height of x, denoted ht(x, T ), is the order type of the set {y T: y < x}. For each ordinal a, the a-th level of T, denoted Leva(T ), is the set {x T: ht(x, T ) = a}. The height ht(T ) of T is the least a such that Leva(T ) =  . A sub-tree of T is a subset S  T with the induced order such that whenever x S and y < x, then y S.
trivial
The smallest, simplest, and usually least interesting example of any object or construction. Every field has a specific definition of what is considered the trivial object of study in that field. The following entries provide examples.
Logic: A conclusion is trivial if it is so obvious that no proof or demonstration is required to establish its truth.
trivial automorphism
The identity function is a trivial automorphism of any set.
Turing machine
An abstract machine consisting of a collection of ordered quadruples, corresponding to i) the contents of the current memory location; ii) the current state of the machine; iii) the action to be performed (erase or write to the current memory location and move to a new memory location); and iv) the new state of the machine. All modern computers are universal Turing machines.
Tychonoff’s Theorem
If {Xa}, a in some index set A, is any family of compact topological spaces, then the cartesian product of the Xa, with the product topology, is compact.
ultrafilter
A filter F on a set X is called an ultrafilter if for every subset Y of X, either Y is in F or the complement of Y is in F.
uncountable
A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”).
Related MiniText: Infinity -- You Can't Get There From Here...
uncountably infinite
See uncountable.
union
The union of two sets A and B is the set consisting of those elements which are in A or in B or in both, and is denoted by
If the union is taken over a family of sets {Ai}i = 1, 2, ..., n, then it is the set consisting of those elements that are in at least one of the sets in the collection.
Sometimes the word “union” is used to indicate the sumset of a set A, and is then loosely described as “union A.” In this latter case the union is understood to be the union over the elements of A, and is denoted by
When considering an algebra of sets, the union of two or more sets is sometimes called the join of the sets.
Cf. intersection.
unit circle
A circle with radius 1.
unit interval
The interval on the real number line from 0 to 1, inclusive.
unit square
The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary.
universal quantifier
See predicate calculus.
universal Turing machine
A Turing machine capable of performing, given the appropriate program, the actions of any other Turing machine.
unordered pair
See flat pair.
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