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
and 
are two binary relations, then their composition 
is the relation
is defined by the rule that says 
if and only if there is an element 
such that 
(i.e. 
and 
).
In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. 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
and 
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:
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 are sometimes regarded as the morphisms 
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
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
and are two binary relations, then their composition is the relation
is defined by the rule that says if and only if there is an element such that (i.e. and ).In particular fields, authors might denote by R ∘ S what is defined here to be S ∘ R. 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
and 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:
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
are sometimes regarded as the morphisms 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
Comments
Post a Comment