Difference between revisions of "Binary operation"
From Online Dictionary of Crystallography
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| − | <font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color=" | + | <font color="blue">Loi de composition</font> (''Fr''). <font color="red">Zweistellige Verknüpfung</font> (''Ge''). <font color="black">Operazione binaria</font> (''It''). <font color="purple">二項演算</font> (''Ja''). <font color="brown">Бинарная операция</font> (''Ru''). <font color="green">Operación binaria</font> (''Sp''). |
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'''. | 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 | + | 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(@) | + | 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'' | + | 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]] | ||
Latest revision as of 17:03, 28 November 2017
Loi de composition (Fr). Zweistellige Verknüpfung (Ge). Operazione binaria (It). 二項演算 (Ja). Бинарная операция (Ru). Operación binaria (Sp).
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.