Define group. Show that the set of all integers Z forms group under addition operation.

This answer is restricted. Please login to view the answer of this question.

Login Now

A group is a non empty set G equivalent with a binary operation *. It is denoted by (G, *)

G × G → G satisfying the following axion

  1. Closure: ∀ a,b ∈ G,   a * b ∈ G
  2. Associatie: ∀ a,b,c ∈ G(a * b) * c = a * (b * c)
  3. Identity Element: ∀ a ∈ G, ∃ an element e ∈ G such that
    a * e = a = e * a
  4. Inverse Element: ∀ a ∈ G, ∃ a-1 ∈ G such thata * a-1 = e = a-1 * a

To show, set of all integerr Z froms group under addition. Let * be defined by

a * b = a  + b

Now,

1) Closure:

Let a, b ∈ Z

then, a * b = a + b ∈ Z  so it is closed

2) Associative:

Let a, b, c ∈ Z then

(a * b) * c = (a + b) * c = a + b + c

a * (b * c) = a * (b + c) = a + b + c

so, it is associative

3) Identity Element:

Here, O ∈ Z such that

a * 0 = a + 0 = a

0 * a = 0  + a = a

Identity Element = e = 0

4) Ezesistence of inverse:

Here, ∀ a ∈ Z there exists -a ∈ Z so,

a * -a = a – a = e

-a * a = -a + a = 0 = e

so, inverse element = -a

Hence, all conditions are satisfied. So, Z forms group under addition operation.

If you found any type of error on the answer then please mention on the comment or report an answer or submit your new answer.
Leave your Answer:

Click here to submit your answer.

Discussion
0 Comments
  Loading . . .