Y= N/D D≠0
Hence if
denominator is zero then real number can never be possible.
So 1/0, 2/0, 3/0,………..are practically
impossible.
i.e. 1/0=
undefined.
Archimedean Property of real numbers
Theorem:- Let a be any real number and b any +ve real number. Then there exists a
positive integer n such that
nb>a
Proof:- Given that aϵ R & bϵ R+
So there are two possible cases.
Case 1:-
When a≤0
In
this case the relation nb>a is always true because the value of nb is always
+ve.
Case 2:- When a>0
Let
us that there exists no +ve integar such that nb>a
Then
we have nb≤a ⩝ n∈N
It means
that a be the upper bound of set S which is given by
S= {b,2b,3b,………..} = {nb:n∈N}.
Boundness of subsets of R
Upper bound of a subset
of R:-
Let S be a subset of real numbers. If there exist a real
number K , such that x≤K ⩝ x∈S
Then K is called an upper bound of set S.
If there
exists an upper bound for a set S then it is called “bounded above”.
Example:-
The set S= {………-4,-3,-2,-1} is bounded above & 9 is an upper bound.
Addition & Multiplication of Modulo - Abstract Algebra
Addition Modulo
Now we are going to discuss a new type of addition which is known as “addition modulo m” and written in the form
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
Properties of Group
- The identity element of a group is always unique.
- The inverse of each element of a group is unique, i.e., in a group G with operation * for every
, there is only element
such that
, e being the identity.
Table for Group- Abstract Algebra
The composition tables are useful in examining the following axioms in the manner explained below:
- Closure Axiom : If all the elements of the table belong to the set G (say) then G is closed under the Composition a (say). If any of the elements of the table does not belong to the set, the set is not closed.
Composition Table - Abstract Algebra
A Binary Operation in a finite set can completely be described with the help of a table. This table is well known as composition table. The composition table helps us to verify most of the properties satisfied by the binary operations. This table can be formed as follows:
Order of a Group- Abstract Algebra
Finite and infinite Groups:
If a group contains a finite number of distinct elements, it is called finite group otherwise an infinite group.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite. Commutative Group or Abelian Group
If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group
is said to be an Abelian group or commutative group if
, 
.
A group which is not Abelian is called a non-Abelian group. The group 
is called the group under addition while the group 
is known as group under multiplication.
is called the group under addition while the group is known as group under multiplication.
Algebraic Structure- Abstract Algebra
A non-empty set 
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “
” is a binary operation on
, then 
is an algebraic structure.
are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “” is a binary operation on, then is an algebraic structure.are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.Identity and Inverse
Identity: A composition 
in a set
is said to possesses of an identity if there exists an element 
such that
Moreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to
.
in a set is said to possesses of an identity if there exists an element such thatMoreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to. Binary Operations
The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers m,n another natural number m+n, similarly the multiplication operation gives for the pair m,n the number m,n in N again. These types of operations arc found to exist in many other sets. Thus we give the following definition.
Archimedean Property of Real Number
Let be any real number. Then there exists a natural number such that .
This theorem is known as the Archimedean property of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).
Vector Space (Theorem-5)
Statement:- The necessary & sufficient conditions for a non-empty subset W of a
vector spaceV(F) to be a subspace of V are
(
1) α∊W, β∊W ⇨ α-β∊W
(2) a∊F, α∊W ⇨ aα∊W
Proof:- Necessary Condition:-
Let V be
a vector space over the field F and W be its subspace.
∴
W will be a vector space over the same field F.
Field & Integral Domain
Integral domain:-
1-
A commutative ring with unity element and without zero divisors is called an
integral domain.
2-
An algebraic system (D,+,.) where D is a non-empty set with two binary
compositions to be denoted by addition and multiplication is called an integral
domain if following axioms are satisfied-
Definition of Ring
Ring:-
An algebraic system (R,+,.) consisting of a
non-empty set R with two binary composition (to be denoted by addition and
multiplication) is called a ring if following axiom are satisfied-
(R,+) is
an abelian group.
1- Closure axiom
2- Associative law
3- Identity element
4- Inverse axiom
(R,.) is
a semi group.
1- Closure axiom
2- Associative law
3- Multiplication distributive over
addition i.e.
a.(b+c)=a.b+a.c, ∀ a, b, c∊R
and (b+c).a=b.a+c.a, ∀a, b, c∊R
Linear Transformation Theorem-1
Statement:-
Let T be
a linear transformation from avector space U into V over the field F.Then T is
non-singular iff T is one-one.
Proof:-
Let T be
a non-singular transformation from U into V.
Let α1,α2∊U such that
Vector Space Question-1
Question:- Express (1,2,3) as a linear combination of (1,1,1),(2,-1,1) and (1,-2,5)
in V3(R).
Solution:- Let a1,a2,a3∊R such that
(1,2,3)=a1(1,1,1)+a2(2,-1,1)+a3(1,-2,5)……………………(a)
(1,2,3)=(a1,a1,a1)+(2a2,-a2,a2)+(a3,-2a3,5a3)
Definition of Group
An
algebraic system (G,o) where G be a non-empty set with o as defined binary
operation is called a group if following axiom are satisfied-
1- Closure Axiom:-
If a, b∊G ⇨ aob∊G , ∀ a,b∊G
then G
is said to be closed under the binary operation.
Binary Operation & Algebric Structure
Binary operation or Binary composition on a set:-
Let G be
a non-empty set then an operation ‘o’ on the non-empty set G is called binary
operation.
If a∊G, b∊G
⇒ aob∊G, ∀ a, b∊G
This
property is called closure property and if this is satisfied then G is said to
be closed under the binary composition ‘o’.
Subscribe to:
Posts (Atom)