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Showing posts from October, 2013

Group Homomorphism

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Definition :-
 a)   Homomorphism onto :-   A mapping    f   from a group  G  onto  G'  is called homomorphism of   G'  , if an only if   a , b  is element of   G  .

                                     f ( a , b ) = f (a) . f(b)

  In this case group  G'  is said to be homomorphic image of group   G  or else the group  G  is said to be homomorphic to group   G'  .

b)   Homomorphism into :-  A mapping  f  from a group  G  into a grroup  G'  is called homomorphism of  g  if and only if   a,  b  is element of  G   .

                                    f ( a , b ) = f (a) f(b)   .

In this case   G"  will not be said to be homomorphic image of  G  which will be   f(G)  to be a subgroup of  G'   .

c)   Endomorphism :-   A homomorphism of a group into itself is called an Endomorphism .

Types of homomorphic maps If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the g…

Binary Composition

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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
In other words, 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…

Rings

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Here in this post we shall deal with an algebraic structure equipped with two binary  compositions denoted additively and multiplicatively i.e. by +  and   .   and it will be known as Ring .

Definition :-  A non empty set R with two binary compositions to be denoted additively and multiplicatively by symbol  +  and  .  is called a Ring  ( R, + , . )  if it satisfies the following axioms :-

R1 The set R is an abelian group for the additive composition.
R2  Multiplication is binary composition which is associative .
R3  Multiplication is both right  and left distributive with regards to addition .

 So we can understand the Ring as follows

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers
. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . There are familiar properties for multiplication and addition of the integers. These properties serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · called additio…

Convergent Series

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Convergent Series :-
 "If   {Sn}  tends to a finite and definite system   S,  then   S  is defined to be the sum to infinity of the series and the series is said to be Convergent to the sum  S"  .  "Convergent Series" redirects here. For the short story collection, see Convergent Series (short story collection). In mathematics, a series is the sum of the terms of a sequence of numbers.
Given a sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,
A series is convergent if the sequence of its partial sums converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a limit such that for any arbitrarily small positive number , there is a large integer such that for all ,
A series that is not convergent is said to be divergent.

 Some Examples of Convergent and Divergent series :-
The reciprocals of the positive integers produce a divergent series (harmonic series): Alternatin…