In college algebra there are mainly four topics namely number theory ,groups  matrices and determinants groups and vectors. In this section we shall throw some light on the topic called groups .Group theory is some what related to Sets theory expect the laws in the group theory.

College Algebra Homework Help

In this section first we shall define a group
Definition of a group : A non empty set S which satisfies the four axioms namely
1.Closed axiom
2.Associative axiom
3.Inverse axiom
4.Identity axiom

If the above four axioms are satisfied then we say that the set S forms a group.
Definition of a abelian group : A non empty set S is said to form an abelian group if it satisfies all the above four axioms and the Commutative law
We shall discuss each axiom in detail with an example
Example1
1.Closed axiom : Suppose there is a set S= { 1,2,3,4.........} Considering S as the set of Natural numbers we will define a binary operation with two elements namely  a and b  under multiplication given by a*b.Here * is taken as Multiplication
Now if you consider any two elements from the set S that is 5 and 7,the operation is a*b that is 5*7=35 this belongs to set S of Natural numbers Hence closed axiom is satisfied.
2.Associative axiom : The Associative axiom is given by (a*b)*c=a*(b*c).
Consider three elements such as 8,2,6 now let us check the validity of this law:
Let a=8 b=2, c=6
Consider Left hand side equation
(a*b)*c= (8*2)*6=16*6=96.---------- (1)
Consider Right hand side equation
a*(b*c)=8*(2*6)=8*12=96------------ (2)
So from (1) and (2) we Validate Associative law is satisfied.
3.Identity axiom :
The Identity axiom is given by a*e=a
Here e is the identity element now if we multiply any element of S we should get back the same element, that means the element satisfying this condition is 1 .Consider a=6
6*1=6. Hence  1 is the identity element.
4 Inverse axiom : The inverse axiom is given by a*a-1= e
Here a-1is called the identity element .
If we try to multiply 1/3 to 3 we get the identity Therefore if we consider a=3 then a-1is 1/3 which gives 1.
3*1/3=1.
Hence inverse axiom is satisfied.
Thus we conclude that The set S forms a group under multiplication.
Example 2
1.Closed axiom : Suppose there is a set M= { 1,2,3,4.........} Considering S as the set of Natural numbers we will define a binary operation with two elements namely  a and b  under addition given by a*b.Here * is taken as addition
Now if you consider any two elements from the set S that is 7and8 ,the operation is a*b that is 7*8=14 this belongs to set S of Natural numbers Hence closed axiom is satisfied.
2.Associative axiom : The Associative axiom is given by (a*b)*c=a*(b*c).
Consider three elements such as 8,2,6 now let us check the validity of this law:
Let a=8 b=2, c=6
Consider Left hand side equation
(a*b)*c= (8*2)*6=16.---------- (1)
Consider Right hand side equation
a*(b*c)=8*(2*6)=8*8=16------------ (2)
So from (1) and (2) we Validate Associative law is satisfied.
3.Identity axiom : The Identity axiom is given by a*e=a
Here e is the identity element now if we multiply any element of S we should get back the same element, that means the element satisfying this condition is 1 .Consider a=9
9*0=9Hence  0 is the identity element.
4 Inverse axiom : The inverse axiom is given by a*a-1= e
Here a-1is called the identity element .
If we try to multiply 1/3 to 3 we get the identity Therefore if we consider a=3 then a-1is 1/3 which gives 1.
3*0=3.
Hence inverse axiom is satisfied.
Thus we conclude that The set S forms a group under addition