|
Rule Of Product
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there are ways of doing something and ways of doing another thing, then there are ways of performing both actions.Johnston, William, and Alex McAllister. ''A transition to advanced mathematics''. Oxford Univ. Press, 2009. Section 5.1 Examples : \begin & \underbrace & & \underbrace \\ \mathrm\ \mathrm\ \mathrm\ \mathrm & \mathrm & \mathrm\ \mathrm\ \mathrm & \mathrm \end : \begin \mathrm\ \mathrm\ \mathrm\ \mathrm\ \mathrm & \mathrm. \\ & \overbrace \end In this example, the rule says: multiply 3 by 2, getting 6. The sets and in this example are disjoint sets, but that is not necessary. The number of ways to choose a member of , and then to do so again, in effect choosing an ordered pair each of whose components are in , is 3 × 3 = 9. As another exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Principles
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for Enumerative combinatorics, enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same cardinality, number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a Discrete mathematics, discrete context. Many combinatorial identities arise from Double counting (proof technique), double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Rule of sum The rule of sum is an intuitive principle stating that if there are ''a'' possible outcomes for an event (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiply
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a '' product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 times 4" and evaluated as 4+4+4, where 3 is the multiplier, but a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theory), 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 family of sets, families of sets and to indexed family, indexed families of sets. By definition, a collection of sets is called a ''family of sets'' (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An \left(A_i\right)_, is by definition a set-valued Function (mathematics), function (that is, it is a function that assigns a set A_i to every ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unordered pair'', denoted , always 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] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . 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 sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -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. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Sum
In combinatorics, the addition principle or rule of sum is a basic counting principle. Stated simply, it is the intuitive idea that if we have ''A'' number of ways of doing something and ''B'' number of ways of doing another thing and we can not do both at the same time, then there are A + B ways to choose one of the actions. In mathematical terms, the addition principle states that, for disjoint sets ''A'' and ''B'', we have , A\cup B, = , A, + , B, , provided that the intersection of the sets is without any elements. The rule of sum is a fact about set theory, as can be seen with the previously mentioned equation for the union of disjoint sets A and B being equal to , A, + , B, . The addition principle can be extended to several sets. If S_1, S_2,\ldots, S_n are pairwise disjoint sets, then we have:, S_1, +, S_2, +\cdots+, S_, = , S_1 \cup S_2 \cup \cdots \cup S_n, . This statement can be proven from the addition principle by induction on ''n''. Simple example A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Principles
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for Enumerative combinatorics, enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same cardinality, number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a Discrete mathematics, discrete context. Many combinatorial identities arise from Double counting (proof technique), double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Rule of sum The rule of sum is an intuitive principle stating that if there are ''a'' possible outcomes for an event (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
