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Difference between revisions of "Binary operation"

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#REDIRECT [[Algebraic operation]]
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An [[algebraic operation]] on a set $A$ with two operands in a given order, hence a function from $A\times A \rightarrow A$.  Such an operator may be written in conventional function or prefix form, as $f(a,b)$, occasionally in postfix form, as $a\,b\,\omega$ or $(a,b)\omega$, but more commonly in infix form as $a \star b$ where $\star$ is the operator symbol.  Many arithmetic, algebraic and logical functions are expressed as binary operations, such as [[addition]], [[subtraction]], [[multiplication]] and [[division]] of various classes of numbers; [[conjunction]], [[disjunction]] and [[implication]] of propositions.
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A binary operation is ''partial'' if it is not defined on all pairs $(a,b) \in A \times A$ (as for example division by zero is not defined).  Properties of binary operations which occur in many contexts include
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* [[Commutativity]]: $a \star b = b \star a$;
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* [[Associativity]]: $a \star (b \star c) = (a \star b) \star c$;
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* [[Idempotence]]: $a \star a = a$.
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,  "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. '''20''' Springer  (1958) {{ZBL|0081.01704}}</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 19:47, 13 November 2016

An algebraic operation on a set $A$ with two operands in a given order, hence a function from $A\times A \rightarrow A$. Such an operator may be written in conventional function or prefix form, as $f(a,b)$, occasionally in postfix form, as $a\,b\,\omega$ or $(a,b)\omega$, but more commonly in infix form as $a \star b$ where $\star$ is the operator symbol. Many arithmetic, algebraic and logical functions are expressed as binary operations, such as addition, subtraction, multiplication and division of various classes of numbers; conjunction, disjunction and implication of propositions.

A binary operation is partial if it is not defined on all pairs $(a,b) \in A \times A$ (as for example division by zero is not defined). Properties of binary operations which occur in many contexts include


References

[a1] R.H. Bruck, "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704
How to Cite This Entry:
Binary operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_operation&oldid=34697