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triangle – vertex set
triangle
Geometry: A closed plane figure with three straight sides meeting at three vertices. If one side of a triangle is chosen as the base, then the height of the triangle is the perpendicular distance to the base from the vertex opposite the base.
Triangles are classified by their angle measures:- Acute – all angles less than 90°.
- Obtuse – one angle greater than 90°.
- Right – one angle exactly 90°.
- Scalene – all angles and sides unequal.
- Isosceles – two angles equal (equivalently, two sides equal).
- Equilateral – all angles equal (equivalently, all sides equal).
On a right triangle, the sides adjacent to the right angle are called the legs, and the side opposite is called the hypotenuse.
 Cf. Pythagorean Theorem.
Graph Theory: A cycle with three vertices.
trigonometric function
The “trig” functions are transcendental functions defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions. See the related article for a complete description.
Related article: Trig Functions and Identities
trigonometry
The mathematical subject concerned with trigonometric functions, which are defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions.
Related article: Trig Functions and Identities
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.
tuple
An ordered list of elements from a set, usually represented as (a1, a2, a3, ... ,an). (Sometimes angle brackets are used in place of parentheses.) A tuple with only two elements is called an ordered pair, and tuples with 3, 4, and 5 elements are called ordered triples, quadruples, and quintuples, respectively. In general, a tuple with n elements is called an n-tuple. A relation on a family of sets is represented by a set of tuples.
Cf. flat pair.
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.
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.
upper bound
An upper bound of a set with an order relation (such as “ < ”) defined on it is an element which is greater than or equal to every element in the set.
Cf. least upper bound.
vector
A quantity having two components; a magnitude component and a direction component. In n-dimensional Euclidean space, a vector is representable by an ordered tuple (a1, a2, a3, ... an) whose elements are called the components of the vector. In this case the magnitude of the vector is given by the square root of the sum of the squares of the components of the vector.
vertex
Geometry: In a plane figure, a point which is a common end-point for two or more lines or curves.
Graph Theory: One of two kinds of entities in a graph.
Cf. edge.
vertex set
The set of vertices of some graph. For a graph G, the vertex set of G is denoted by V(G), or, if there is no ambiguity as to the graph in question, simply by V.
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