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Addition Principle
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] |
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] |
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] |
Mathematical Induction
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. The binomial coefficients are the numbers that appear in Pascal's triangle. Pascal's rule states that for positive integers ''n'' and ''k'', + = , where \tbinom is the binomial coefficient, namely the coefficient of the term in the expansion of . There is no restriction on the relative sizes of and ; in particular, the above identity remains valid when since \tbinom = 0 whenever . Together with the boundary conditions \tbinom = \tbinom= 1 for all nonnegative integers ''n'', Pascal's rule determines that \binom = \frac, for all integers . In this sense, Pascal's rule is the recurrence relation that defines the binomial coefficients. Pascal's rule can also be generalized to apply to multinomial coefficients. Combinatorial proof Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof. ''Proof''. Recall that \tbinom e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication Principle
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 example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Principle
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 purposes. Bijective proofs are utilized to demonstrate that two sets have the same 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 context. Many combinatorial identities arise from 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 ways to do something) and ''b'' possible outcomes for another event (or ways to do another th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
