In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set; some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be called a bifurcating arborescence—a term which appears in some old programming books, before the modern computer science terminology prevailed, it is possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special case of an ordered K-ary tree, where k is 2.
In mathematics, what is termed binary tree can vary from author to author. Some use the definition used in computer science, but others define it as every non-leaf having two children and don't order the children either. In computing, binary trees are used in two different ways: First, as a means of accessing nodes based on some value or label associated with each node. Binary trees labelled this way are used to implement binary search trees and binary heaps, are used for efficient searching and sorting; the designation of non-root nodes as left or right child when there is only one child present matters in some of these applications, in particular it is significant in binary search trees. However, the arrangement of particular nodes into the tree is not part of the conceptual information. For example, in a normal binary search tree the placement of nodes depends entirely on the order in which they were added, can be re-arranged without changing the meaning. Second, as a representation of data with a relevant bifurcating structure.
In such cases the particular arrangement of nodes under and/or to the left or right of other nodes is part of the information. Common examples occur with Huffman coding and cladograms; the everyday division of documents into chapters, paragraphs, so on is an analogous example with n-ary rather than binary trees. To define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an extended binary tree is needed for that purpose. An extended binary tree is thus recursively defined as: the empty set is an extended binary tree if T1 and T2 are extended binary trees denote by T1 • T2 the extended binary tree obtained by adding a root r connected to the left to T1 and to the right to T2 by adding edges when these sub-trees are non-empty. Another way of imagining this construction is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles. A binary tree is a rooted tree, an ordered tree in which every node has at most two children.
A rooted tree imparts a notion of levels, thus for every node a notion of children may be defined as the nodes connected to it a level below. Ordering of these children makes possible to distinguish left child from right child, but this still doesn't distinguish between a node with left but not a right child from a one with right but no left child. The necessary distinction can be made by first partitioning the edges, i.e. defining the binary tree as triplet, where is a rooted tree and E1 ∩ E2 is empty, requiring that for all j ∈ every node has at most one Ej child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees. Tree terminology so varies in the literature. A rooted binary tree has a root node and every node has at most two children. A full binary tree is a tree in which every node has 2 children. Another way of defining a full binary tree is a recursive definition.
A full binary tree is either:A single vertex. A tree whose root note has two subtrees. In a complete binary tree every level, except the last, is filled, all nodes in the last level are as far left as possible, it can have between 2h nodes at the last level h. An alternative definition is a perfect tree; some authors use the term complete to refer instead to a perfect binary tree as defined above, in which case they call this type of tree an complete binary tree or nearly complete binary tree. A complete binary tree can be efficiently represented using an array. A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level. An example of a perfect binary tree is the ancestry chart of a person to a given depth, as each person has two biological parents. In the infinite complete binary tree, every node has two children; the set of all nodes is countably infinite, but the set of all infinite paths from the root
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.
Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.
There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.
For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. The result of a subtraction is called a difference. Subtraction is signified by the minus sign. For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, a total of 3 apples. Therefore, the difference of 5 and 2 is 3, that is, 5 − 2 = 3. Subtraction represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, irrational numbers, decimals and matrices. Subtraction follows several important patterns, it is anticommutative. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters; because 0 is the additive identity, subtraction of it does not change a number. Subtraction obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond.
General binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks. Subtraction of small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. In advanced algebra and in computer algebra, an expression involving subtraction like A − B is treated as a shorthand notation for the addition A +. Thus, A − B contains two terms, namely A and −B; this allows an easier use of commutativity. Subtraction is written using the minus sign "−" between the terms; the result is expressed with an equals sign. For example, 2 − 1 = 1 4 − 2 = 2 6 − 3 = 3 4 − 6 = − 2 There are situations where subtraction is "understood" though no symbol appears: A column of two numbers, with the lower number in red indicates that the lower number in the column is to be subtracted, with the difference written below, under a line.
This is most common in accounting. Formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend; the result is the difference. All of this terminology derives from Latin. "Subtraction" is an English word derived from the Latin verb subtrahere, in turn a compound of sub "from under" and trahere "to pull". Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted". From minuere "to reduce or diminish", one gets "minuend", "thing to be diminished". Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c; this movement to the right is modeled mathematically by addition: a + b = c. From c, it takes b steps to the left to get back to a; this movement to the left is modeled by subtraction: c − b = a. Now, a line segment labeled with the numbers 1, 2, 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3, it takes 2 steps to the left to get to position 1, so 3 − 2 = 1.
This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number. From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0, but 3 − 4 is still invalid. The natural numbers are not a useful context for subtraction; the solution is to consider the integer number line. From 3, it takes 4 steps to the left to get to −1: 3 − 4 = −1. Subtraction of natural numbers is not closed; the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number; such a case uses one of two approaches: Say that 26 cannot be subtracted from 11. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15. Subtraction of real numbers is defined as addition of signed numbers. A number is subtracted by adding its additive inverse.
We have 3 − π = 3 +. This helps to keep the ring of real numbers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it. A ring has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything. Subtraction is anti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers a − b = −. Subtraction is non-associative. Should the expres
Addition is one of the four basic operations of arithmetic. The addition of two whole numbers is the total amount of those values combined. For example, in the adjacent picture, there is a combination of three apples and two apples together, making a total of five apples; this observation is equivalent to the mathematical expression "3 + 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting items, addition can be defined on other types of numbers, such as integers, real numbers and complex numbers; this is part of a branch of mathematics. In algebra, another area of mathematics, addition can be performed on abstract objects such as vectors and matrices. Addition has several important properties, it is commutative, meaning that order does not matter, it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 1 is the same as counting. Addition obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks. Addition of small numbers is accessible to toddlers. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign "+" between the terms; the result is expressed with an equals sign. For example, 1 + 1 = 2 2 + 2 = 4 1 + 2 = 3 5 + 4 + 2 = 11 3 + 3 + 3 + 3 = 12 There are situations where addition is "understood" though no symbol appears: A whole number followed by a fraction indicates the sum of the two, called a mixed number. For example, 3½ = 3 + ½ = 3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead; the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration.
For example, ∑ k = 1 5 k 2 = 1 2 + 2 2 + 3 2 + 4 2 + 5 2 = 55. The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands; this is to be distinguished from factors. Some authors call. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is used, both terms are called addends. All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give". Using the gerundive suffix -nd results in "addend", "thing to be added". From augere "to increase", one gets "augend", "thing to be increased". "Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was higher than the addends.
Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus. The Middle English terms "adden" and "adding" were popularized by Chaucer; the plus sign "+" is an abbreviation of the Latin word et, meaning "and". It appears in mathematical works dating back to at least 1489. Addition is used to model many physical processes. For the simple case of adding natural numbers, there are many possible interpretations and more visual representations; the most fundamental interpretation of addition lies in combining sets: When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is easy to visualize, with little danger of ambiguity, it is useful in higher mathematics. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. One possible fix is to consider collections of objects that can be divided, such as pies or, still bet
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra and functional programming; the term was introduced by Benjamin Peirce in the context of elements of algebras that remain invariant when raised to a positive integer power, means " the same power", from idem + potence. An element x of a magma is said to be idempotent if: x • x = x. If all elements are idempotent with respect to • • is called idempotent; the formula ∀x, x • x = x is called the idempotency law for •. The natural number 1 is an idempotent element with respect to multiplication, so is 0, but no other natural number is. For the latter reason, multiplication of natural numbers is not an idempotent operation. More formally, in the monoid, idempotent elements are just 0 and 1. In a magma, an identity element e or an absorbing element a, if it exists, is idempotent.
Indeed, e • e = e and a • a = a. In a group, the identity element e is the only idempotent element. Indeed, if x is an element of G such that x • x = x x • x = x • e and x = e by multiplying on the left by the inverse element of x. Taking the intersection x∩y of two sets x and y is an idempotent operation, since x∩x always equals x; this means that the idempotency law ∀ x ∩ x = x is true. Taking the union of two sets is an idempotent operation. Formally, in the monoids and of the power set of the set E with the set union ∪ and set intersection ∩ all elements are idempotent. In the monoids and of the Boolean domain with the logical disjunction ∨ and the logical conjunction ∧ all elements are idempotent. In a Boolean ring, multiplication is idempotent. In the monoid of the functions from a set E to a subset F of E with the function composition ∘, idempotent elements are the functions f: E → F such that f ∘ f = f, in other words such that for all x in E, f = f. For example: Taking the absolute value abs of an integer number x is an idempotent function for the following reason: abs = abs is true for each integer number x.
This means that abs ∘ abs = abs holds, that is, abs is an idempotent element in the set of all functions with respect to function composition. Therefore, abs satisfies the above definition of an idempotent function. Other examples include: the identity function is idempotent. If the set E has n elements, we can partition it into k chosen fixed points and n − k non-fixed points under f, kn−k is the number of different idempotent functions. Hence, taking into account all possible partitions, ∑ k = 0 n k n − k is the total number of possible idempotent functions on the set; the integer sequence of the number of idempotent functions as given by the sum above for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, … starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, …. Neither the property of being idempotent nor that of being not is preserved under function composition; as an example for the former, f = x mod 3 and g = max are both idempotent, but f ∘ g is not, although g ∘ f happens to be. As an example for the latter, the negation function ¬ on the Boolean domain is not idempotent, but ¬ ∘ ¬ is.
Unary negation − of real numbers is not idempotent, but − ∘ − is. In computer science, the term idempotence may have a different meaning depending on the context in which it is applied: in imperative programming, a subroutine with side effects is idempotent if the system state remains the same after one or several calls, in other words if the function from the system state space to itself associated to the subroutine is idempotent in the mathematical sense given in the definition; this is a useful property in many situations, as it means that an operation can be repeated or retried as as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was performed or not. A function looking up a customer's name and address in a database is idempotent, since this will not cause the database to change. Changing a customer's address is idempotent, because the final address will be the same no matter how many times it is submitted.
However, placing an order for a cart for the customer is not idempotent, since running the call several t