Difference between revisions of "Binary operation"
From Online Dictionary of Crystallography
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| − | A '''binary operation''' on a set S is a | + | A '''binary operation''' on a set ''S'' is a [[mapping]] ''f'' from the [[Cartesian product]] ''S × S'' to ''S''. A mapping from ''K x S'' to ''S'', where ''K'' need not be ''S'', is called an '''external binary operation'''. |
| − | Many binary operations are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions | + | Many binary operations are commutative (i.e. ''f(a,b) = f(b,a)'' holds for all ''a, b'' in ''S'') or associative (i.e. ''f(f(a,b), c) = f(a, f(b,c))'' holds for all ''a,b,c'' in ''S''). Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations. |
| − | Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), | + | Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@), and composition. |
| − | Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form ''f | + | Binary operations are often written using infix notation such as ''a * b'', ''a + b'', or ''a · b'' rather than by functional notation of the form ''f(a,b)''. Sometimes they are even written just by concatenation: ''ab''. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 14:02, 1 April 2009
Loi de composition (Fr). Zweistellige Verknüpfung (Ge). Operación Binaria (Sp). Бинарная операция (Ru). Operazione binaria (It). 二項演算 (Ja).
A binary operation on a set S is a mapping f from the Cartesian product S × S to S. A mapping from K x S to S, where K need not be S, is called an external binary operation.
Many binary operations are commutative (i.e. f(a,b) = f(b,a) holds for all a, b in S) or associative (i.e. f(f(a,b), c) = f(a, f(b,c)) holds for all a,b,c in S). Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.
Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@), and composition.
Binary operations are often written using infix notation such as a * b, a + b, or a · b rather than by functional notation of the form f(a,b). Sometimes they are even written just by concatenation: ab.