Question

Prove that the set of integers is an abelian group under addition 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

Given:

The set of integers is,

$\[I = \left\{ {.......... - 3, - 2, - 1,0,1,2,3...........} \right\}\]$

The binary operation is '$+$'.

To Prove:

The set of integers is an abelian group under the addition operation i.e., $\[\left( {I, + } \right)\]$ is an abelian group.

Solution:

To prove that $\[\left( {I, + } \right)\]$ 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:

For all $\[a,b \in I \Rightarrow a + b \in I\]$.

Since the sum of two integers is also an integer, so closure property holds. For example: $\[4, - 1 \in I \Rightarrow 4 + \left( { - 1} \right) = 3 \in I\]$.

2. Associative Property:

$\[\left( {a + b} \right) + c = a + \left( {b + c} \right)\,\forall \,a,b,c \in I\]$

Let $\[\,a,b,c \in I\]$ where  $\[a = 1,\,b = - 3,\,c = 4\]$.

Then,

$\[\begin{array}{c}\left( {a + b} \right) + c = \left( {1 + \left( { - 3} \right)} \right) + 4\\ = 2\end{array}\]$

$\[\begin{array}{c}a + \left( {b + c} \right) = 1 + \left( { - 3 + 4} \right)\\ = 2\end{array}\]$

Therefore, $\[\left( {a + b} \right) + c = a + \left( {b + c} \right)$.

Hence, Associative Property is satisfied.

3. Identity Property:

0 is the additive identity for the group of integers.

$\[a + 0 = 0 + a = a\,\,\forall a \in I,0 \in I\]$

Let $a=\[2 \in I\]$, then $\[2 + 0 = 0 + 2 = 2\]$.

Hence, Identity Property is also satisfied.

4.  Inverse Property:

$\[a + \left( { - a} \right) = 0\,\,\forall a \in I, - a \in I,0 \in I\]$

Let $a=\[12 \in I\]$ then there exists a number $\[ - 12 \in I\]$ such that $\[12 + \left( { - 12} \right) = 0\]$

Hence, Inverse Property is also satisfied.

5. Commutative Property:

$\[a + b = b + a\,\,\forall a,b \in I\]$

Let $\[a,b \in I\]$ where $\[a = 7,\,b = 9\]$.

Then,

$\[\begin{array}{c}a + b = 7 + 9\\ = 16\end{array}\]$

$\[\begin{array}{c}b + a = 9 + 7\\ = 16\end{array}\]$

Therefore, $\[a + b = b + a\,$.

Hence, Commutative Property is also satisfied.

Since all the five properties are satisfied, therefore $\[\left( {I, + } \right)\]$ is an abelian group.

Hence proved that the set of integers is an abelian group under the addition operation.

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