**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 ∈ I2,-3 ∈ I ⇒ -1 ∈ I

Hence Closure Property is satisfied.

**2)** **Associative Property**

( a+ b ) + c = a+( b +c) ∀ a , b , c ∈ I2 ∈ 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 ∈ I5 ∈ 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 ∈ Ia=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 ∈ ILet 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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