Wednesday, 30 October 2013

Group Homomorphism




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 groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes.
If h: GG is a group homomorphism, we call it an endomorphism of G. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to Z/2Z.
An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function.


Homomorphisms of abelian groups

If G and H are abelian (i.e. commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by
(h + k)(u) = h(u) + k(u)    for all u in G.
The commutativity of H is needed to prove that h + k is again a group homomorphism.
The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H,L), then
(h + k) o f = (h o f) + (k o f)   and    g o (h + k) = (g o h) + (g o k).
This shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sum of m copies of Z/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.
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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

Saturday, 26 October 2013

Rings

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 addition and multiplication, that map every pair of elements of R to a unique element of R. These operations must satisfy the following properties called ring axioms (the symbol ⋅ is often omitted and multiplication is just denoted by juxtaposition.), which must be true for all a, b, c in R:
  • R is an Abelian group under addition, meaning:
1. (a + b) + c = a + (b + c) (+ is associative)
2. There is an element 0 in R such that 0 + a = a (0 is the zero element)
3. a + b = b + a (+ is commutative)
4. For each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)
  • Multiplication is associative:
5. (ab) ⋅ c = a ⋅ (bc)
  • Multiplication distributes over addition:
6. a ⋅ (b + c) = (ab) + (ac) (left distributivity)
7. (b + c) ⋅ a = (ba) + (ca) (right distributivity)
For many authors, these seven axioms are all that are required in the definition of a ring (such a structure is also called pseudo-ring, or a rng). For others, the following additional axiom is also required:
  • Multiplicative identity
8. There is an element 1 in R such that a ⋅ 1 = 1 ⋅ a = a
Rings which satisfy all eight of the above axioms are sometimes, for emphasis, referred to as unital rings (also called unitary rings, rings with unity, rings with identity or rings with 1). For example, the set of even integers satisfies the first seven axioms, but it does not have a multiplicative identity, and therefore does not satisfy the eighth axiom.
Regarding which convention should be used, Gardner and Wiegandt argue that if one requires all rings to have a unity, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." This article adopts the convention that, unless otherwise stated, a ring is assumed to be unital.
Although ring addition is commutative, so that a + b = b + a, ring multiplication is not required to be commutative; ab need not equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
Some basic properties of a ring follow immediately from the axioms.
  • The additive identity and the additive inverse are unique.
  • The binomial formula holds for any commuting elements (i.e., xy = yx).

Example: Integers modulo 4

Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows. To avoid possible confusions and to keep the usual notation for the arithmetic operations, we will over-line 0, 1, 2, 3 when considering them in Z4.
  • (Addition) The sum \overline{x} + \overline{y} in Z4 is the remainder of x+y (as an integer) when divided by 4. For example, in Z4 we have \overline{2} + \overline{3} = \overline{1} and \overline{3} + \overline{3} = \overline{2}
  • (Multiplication) The product \overline{x} \cdot \overline{y} in Z4 is the remainder of x\cdot y (as an integer) when divided by 4. For example, in Z4 we have \overline{2} \cdot \overline{3} = \overline{2} and \overline{3} \cdot \overline{3} = \overline{1}.
If x is an integer, the remainder of x when divided by 4 is an element of Z4, and this element is often denoted by "x mod 4", or sometimes \overline{x}, which is coherent with above notation. By checking each axiom, one verifies that Z4 is a ring under these operations. Each axiom follows from the fact that the integers form a ring, and converting the integers to Z4. The additive inverse of any \overline{x} in Z4 is the remainder (-x \mod 4) =\overline{-4}. In other words, we have -\overline{x}=\overline{-x}. For example, in Z4, we have -\overline{3}= \overline{-3} = \overline{1}.
Once one has checked that the ring axioms hold, operations within the ring Z4 become easier to carry out. For example, to compute 3 ⋅ (3 − 1) + 1, one first computes the value within the full set of integers (which is 7), and then converts the result by finding the remainder after dividing by 4, which in this case is 3.

Example: 2-by-2 matrices

Consider the set of 2-by-2 matrices, whose entries are real numbers. This set is written:
\mathcal{M}_2(\mathbb{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \bigg|\  a,b,c,d \in \mathbb{R} \right\}
One can check that with the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms. The element is the multiplicative identity element of the ring. This ring is one of the simplest examples of a non-commutative ring. To see that it is not commutative, consider the following multiplications, which give two matrices A and B such that AB is different from BA:
 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}  \neq  \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}  = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
One can generalize this construction by replacing the set of real numbers with any ring (not necessarily commutative), and instead of using 2-by-2 matrices, one can use square matrices of any fixed size; see matrix ring.

Rings with extra structure

A ring may be viewed as an abelian group (by using the addition operation), with extra structure. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
  • An associative algebra is a ring that is also a vector space over a field K. For instance, the set of n-by-n matrices over the real field R has dimension n2 as a real vector space.
  • A ring R is a topological ring if its set of elements is given a topology which makes the addition map ( + : R\times R \to R\,) and the multiplication map ( \cdot : R\times R \to R\,) to be both continuous as maps between topological spaces (where X × X inherits the product topology or any other product in the category). For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
It may be noted that for ring there is no necessity for the existence of multiplicative identity and inverse and also multiplicative composition to be commutative . 

Wednesday, 23 October 2013

Convergent Series

In mathematics, a series is the sum of the terms of a sequence of numbers.
Given a sequence \left \{ a_1,\ a_2,\ a_3,\dots \right \}, the nth partial sum S_n is the sum of the first n terms of the sequence, that is,
S_n = \sum_{k=1}^n a_k.
A series is convergent if the sequence of its partial sums \left \{ S_1,\ S_2,\ S_3,\dots \right \} converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a limit \ell such that for any arbitrarily small positive number \varepsilon > 0, there is a large integer N such that for all n \ge \ N,
\left | S_n - \ell \right \vert \le \ \varepsilon.
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):
    {1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty.
  • Alternating the signs of the reciprocals of positive integers produces a convergent series:
    {1\over 1} -{1\over 2} + {1\over 3} - {1\over 4} + {1\over 5} \cdots = \ln(2)
  • Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):
    {1 \over 1}-{1 \over 3}+{1 \over 5}-{1 \over 7}+{1 \over 9}-{1 \over 11}+\cdots = {\pi \over 4}.
  • The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
    {1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty.
  • The reciprocals of triangular numbers produce a convergent series:
    {1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots = 2.
  • The reciprocals of factorials produce a convergent series (see e):
    \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24}  + \frac{1}{120} + \cdots = e.
  • The reciprocals of square numbers produce a convergent series (the Basel problem):
    {1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+{1 \over 36}+\cdots = {\pi^2 \over 6}.
  • The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
    {1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots = 2.
  • Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
    {1 \over 1}-{1 \over 2}+{1 \over 4}-{1 \over 8}+{1 \over 16}-{1 \over 32}+\cdots = {2\over3}.
  • The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
    \frac{1}{1} +  \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \cdots = \psi.   

    In the analysis of convergence of a series , the series of positive terms hold an important place . obviously for a series of positive terms the sum of   n  terms   Sn   goes on increasing as more and more terms are added up .  However it does not guarantee that the sum of infinite number will exceed any prescribed finite number . It may happen that the increase in the sum goes on decreasing as more and more terms are added up and ultimately this increase become negligible i.e. to say that sum get closer to a definite number .

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