Question

Question :

Determine whether each of the following given sets is a group with respect to the indicated operation. Show your work.

(b) The set E of all even integers with operation addition.

My work :

So I was working on above part (b)

Following is an extract from the note I've for this topic :

Definition of a Group

Let G be a nonempty set and let the operation ∗ is define on G as follows:
∗ : G × G → G
(x, y) → x ∗ y
for all x, y ∈ G.
Then G is said to be a group with respect to the operation ∗ (or (G, ∗) is a
group) provided the following axioms are hold:

(G1) x∗y ∈ G for all x, y ∈ G. (This property is called the closure property.)

(G2) x ∗ (y ∗ z) = (x ∗ y) ∗ z for all x, y, z ∈ G. (This is called the associative
property.)

(G3) There exists an element e, called the identity element, such that x∗e = e ∗ x = x foa all x ∈ G. (This is known as the existence of the identity element).

(G4) For each x ∈ G, there exists an element called the inverse of x denoted by x^-1, such that
x∗x^−1 = e = x^−1 ∗x.
(This is said to be the existence
of the inverse element ).

I proved G1

And then when it comes G2, I considered three any elements in the set E, x y z

So now I've to prove that x+(y+z) = (x+y)+z

I know since they're even integers we can denote them as x=2a , y=2b , z=2c where a b c are integers.

So with or without that how do we prove their associative property?

In real analysis we've been taught that the set of real numbers is associative under addition.

Since E is an subset of R can we simply say E is associative?
Or how?

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Then when it comes to G3(additive identity), can we simply say x+0 = x and 0+x=x from the associative property we've proved just above?


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And how do we prove G4?

Answer 1

As usual, google is your friend. You can start here:

https://math.stackexchange.com/questions/1916798/proof-that-e-g-integers-with-addition-is-a-group