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N-ary Associativity
In algebra, -ary associativity is a generalization of the associative law to -ary operations. A ternary operation is ternary associative if one has always : (abc)de=a(bcd)e=ab(cde); that is, the operation gives the same result when any three adjacent elements are bracketed inside a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ... of five operands. Similarly, an -ary operation is -ary associative if bracketing any adjacent elements in a sequence of operands do not change the result.. References Properties of binary operations {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Law
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arity
In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. Examples In general, functions or operators with a given arity follow the naming conventions of ''n''-based numeral systems, such as binary and hexadecimal. A Latin prefix is combined with the -ary suffix. For example: * A nullary function takes no arguments. ** Example: f()=2 * A unary function takes one argument. ** Example: f(x)=2x * A binary function takes two arguments. ** Example: f(x,y)=2xy * A ternary function takes three arguments. ** Example: f(x,y,z)=2xyz * An ''n''-ary function takes ''n'' arguments. ** Example: f(x_1, x_2, \ldots, x_n)=2\prod_^n x_i Nullary A constant can be treated as the output of an operation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Operation
In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operator is an operator that takes three arguments as input and returns one output. Examples The function T(a, b, c) = ab + c is an example of a ternary operation on the integers (or on any structure where + and \times are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry. In the Euclidean plane with points ''a'', ''b'', ''c'' referred to an origin, the ternary operation , b, c= a - b + c has been used to define free vectors. Since (''abc'') = ''d'' implies ''b'' – ''a'' = ''c'' – ''d'', the directed line segments ''b'' – ''a'' and ''c'' – ''d'' are equipollent and are associated with the same free vector. Any three points in the plane ''a, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence (mathematics)
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
