Question

Prove that set of integer is an abelian group under adittion operation

Answer 1

Answer:

Step-by-step explanation:

An abelian group is a set with an operation that is closed in that set, is associative, has an identity element, has inverses, and is commutative. Addition is already associative and commutative over the set of all integers, and has an identity and an inverse for each integer .

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Solution :

To prove :

The set of integers is an abelian group under addition operation , ie. (Z,+) is an abelian group .

Proof :

Let a , b , c ∈ Z be any three arbitrary elements .

1) Closure property :

We know that , the sum of two integers is again an integer , ie. a + b ∈ Z ∀ a , b ∈ Z .

→ (Z,+) is closed .

2) Associative property :

We know that , the addition of integers is associative , ie. (a + b) + c = a + (b + c) ∀ a , b , c ∈ Z .

→ (Z,+) is associative .

3) Existence of identity element :

We have 0 ∈ Z such that a + 0 = 0 + a = a ∀ a ∈ Z .

→ 0 ∈ (Z,+) is the identity element .

4) Existence of inverse element :

To each a ∈ Z , we have -a ∈ Z such that a + (-a) = (-a) + a = 0 .

→ -a is the inverse of a ∈ (Z,+) .

5) Commutative property :

We know that , the addition of integers is commutative, ie. a + b = b + a ∀ a , b ∈ Z .

→ (Z,+) is commutative .

Hence , (Z,+) is an abelian group .