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Monday, 28 October 2013

Binary Composition

Definition :- If  G  be a non empty set and   a,  b  are the element of  G  then a composition denoted by   o  such that   a  o  b  of the element of   G  is  called a Binary Composition in the set   G  .

   In other words a Binary Composition in a set   G  is a mapping of   G x G  into    G   which associates to each ordered  pair   ( a  ,  b)   of members  of   G  ,  a member of   G   .

You may understand this by following :-

If R\subseteq X\times Y and S\subseteq Y\times Z are two binary relations, then their composition S\circ R is the relation
S\circ R = \{ (x,z)\in X\times Z\mid \exists y\in Y: (x,y)\in R\land (y,z)\in S \}.
In other words, S\circ R\subseteq X\times Z is defined by the rule that says (x,z)\in S\circ R if and only if there is an element y\in Y such that x\,R\,y\,S\,z (i.e. (x,y)\in R and (y,z)\in S).
In particular fields, authors might denote by RS what is defined here to be SR. The convention chosen here is such that function composition (with the usual notation) is obtained as a special case, when R and S are functional relations. Some authors prefer to write \circ_l and \circ_r explicitly when necessary, depending whether the left or the right relation is the first one applied.
A further variation encountered in computer science is the Z notation: \circ is used to denote the traditional (right) composition, but ⨾ ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition. This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory, as well as the notation for dynamic conjunction within linguistic dynamic semantics.
The binary relations R\subseteq X\times Y are sometimes regarded as the morphisms R\colon X\to Y in a category Rel which has the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. The category Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of this is found in the theory of allegories.

Consider the Set    I   of integers .  The operation of Addition and Multiplication and Subtraction  are all Binary Compositions because

                         a o  b  =  a  +   b     = integer is element of   I 
                         a  o  b =  a  .   b      = integer is  element of   I
                         a  o  b  =  a - b        = integer is   element of  I