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Difference between revisions of "Binary operation"

From Online Dictionary of Crystallography

 
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A '''binary operation''' on a set S is a function ''f'' from the [[Cartesian product]] S × S to S. A binary function from K and S to S, where K need not be S, is called an ''' external binary operation '''.
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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 on a single set.
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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(^), and super-exponentiation(@).
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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 juxtaposition: ab.
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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.