Abelian Group Example
Problem-: Prove that ( I, + ) is an abelian group. i.e. The set of all integers I form an abelian group with respect to binary operation ‘+’.
Solution-:
Set= I ={ ……………..-3, -2 , -1 , 0, 1, 2 , 3……………… }.
Binary Operation= ‘+’
Algebraic Structure= (I ,+)
We have to prove that (I,+) is an abelian group.
To prove that set of integers I is an abelian group we must satisfy the following five properties that is Closure Property, Associative Property, Identity Property, Inverse Property, and Commutative Property.
1) Closure Property
∀ a , b ∈ I ⇒ a + b ∈ I
2,-3 ∈ I ⇒ -1 ∈ I
Hence Closure Property is satisfied.
2) Associative Property
( a+ b ) + c = a+( b +c) ∀ a , b , c ∈ I
2 ∈ I, -6 ∈ I , 8 ∈ I
So, LHS= ( a + b )+c
= (2+ ( -6 ) ) + 8 = 4
RHS= a + ( b + c )
=2 + ( – 6 + 8 ) = 4
Hence RHS = LHS
Associative Property is also Satisfied
3) Identity Property
a + 0 = a ∀ a ∈ I , 0 ∈ I
5 ∈ I
5+0 = 5
-17 ∈ I
-17 + 0 = – 17
Identity property is also satisfied.
4) Inverse Property
a + ( -a ) = 0 ∀ a ∈ I , -a ∈ I ,0 ∈ I
a=18 ∈ I then ∋ a number -18 such that 18 + ( -18 ) = 0
So, Inverse property is also satisfied.
5) Commutative Property
a + b = b + a ∀ a , b ∈ I
Let a=19, b=20
LHS = a + b
= 19+( -20 ) = -1RHS = b + a
= -20 +19 = -1
LSH=RHS
Commutative Property is also satisfied.
We can see that all five property is satisfied. Hence (I,+) is an Abelian Group.
Note-: (I,+) is also Groupoid, Monoid and Semigroup.
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