* picture info Cartesian Product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''- tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards A ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Cartesian Product Qtl1 Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics *Cartesian closed category, a closed category in category theory *Cartesian coordinate system, modern rectangular coordinate system * Cartesian diagram, a construction in category theory *Cartesian geometry, now more commonly called analytic geometry * Cartesian morphism, formalisation of ''pull-back'' operation in category theory *Cartesian oval, a curve *Cartesian product, a direct product of two sets *Cartesian product of graphs, a binary operation on graphs *Cartesian tree, a binary tree in computer science Philosophy *Cartesian anxiety, a hope that studying the world will give us unchangeable knowledge of ourselves and the world *Cartesian circle, a potential mistake in reasoning *Cartesian doubt, a form of methodical skepticism as a basis for philosophical rigor *Cartesian dualism, the philosophy of the distinction between mind and ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Disjoint Sets In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is d ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Relation (mathematics) In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Function (mathematics) In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Axiom Schema Of Specification In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for ''unrestricted'' comprehension, discussed below. Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory. Statement One instance of the schema is included for each formula φ in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in φ. In the formal language of set theory, the axiom schema is: :\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow x \in A \land \varphi(x, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Axiom Of Power Set In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall x \, \exists y \, \forall z \, \in y \iff \forall w \, (w \in z \Rightarrow w \in x)/math> where ''y'' is the Power set of ''x'', \mathcal(x). In English, this says: :Given any set ''x'', there is a set \mathcal(x) such that, given any set ''z'', this set ''z'' is a member of \mathcal(x) if and only if every element of ''z'' is also an element of ''x''. More succinctly: ''for every set x, there is a set \mathcal(x) consisting precisely of the subsets of x.'' Note the subset relation \subseteq is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, \in. By the axiom of extensionality, the set \mathcal(x) is unique. The axiom of power set appears in most axiomatizations of set theory. It i ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Axiom Of Union In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo. The axiom states that for each set ''x'' there is a set ''y'' whose elements are precisely the elements of the elements of ''x''. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall A\, \exists B\, \forall c\, (c \in B \iff \exists D\, (c \in D \land D \in A)\,) or in words: :Given any set ''A'', there is a set ''B'' such that, for any element ''c'', ''c'' is a member of ''B'' if and only if there is a set ''D'' such that ''c'' is a member of ''D'' and ''D'' is a member of ''A''. or, more simply: :For any set A, there is a set \bigcup A\ which consists of just the elements of the elements of that set A. Relation to Pairing The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the axiom of pairing, this implies that for any two sets, the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Axiom Of Pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall A \, \forall B \, \exists C \, \forall D \, \in C \iff (D = A \lor D = B)/math> In words: :Given any object ''A'' and any object ''B'', there is a set ''C'' such that, given any object ''D'', ''D'' is a member of ''C'' if and only if ''D'' is equal to ''A'' or ''D'' is equal to ''B''. Or in simpler words: :Given two objects, there is a set whose members are exactly the two given objects. Consequences As noted, what the axiom is saying is that, given two objects ''A'' and ''B'', we can find a set ''C'' whose members are exactly ''A'' and ''B''. We can use the axiom of extensionality to show that this set ''C'' is unique. We call t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Power Set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as wel ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Ordered Pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In contrast, the unordered pair equals the unordered pair .) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Real Number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]