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?
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
To determine whether the set E of all even integers with the operation of addition is a group, we need to show that it satisfies the four group axioms outlined in your note.
1. Closure Property (G1):
To prove closure, we need to show that for any two even integers x and y, their sum x + y is also an even integer.
Since x and y are even, we can express them as x = 2a and y = 2b, where a and b are integers.
Now, let's calculate the sum: x + y = 2a + 2b = 2(a + b).
Since the sum of two integers is always an integer, the expression 2(a + b) represents an even integer. Therefore, the closure property holds for E under addition.
2. Associative Property (G2):
To prove the associative property, we need to show that for any three even integers x, y, and z, the expression (x + y) + z is equal to x + (y + z).
Let's assume x = 2a, y = 2b, and z = 2c, where a, b, and c are integers.
Now, evaluate both sides of the equation:
(x + y) + z = (2a + 2b) + 2c = 2(a + b) + 2c = 2((a + b) + c)
x + (y + z) = 2a + (2b + 2c) = 2a + 2(b + c) = 2(a + (b + c))
Since addition is associative for integers, we can conclude that (x + y) + z is equal to x + (y + z). Therefore, the associative property holds for E under addition.
3. Existence of Identity Element (G3):
To prove the existence of an identity element, we need to show that there exists an even integer e in E such that x + e = e + x = x for any x in E.
In this case, the identity element is 0, which is an even integer. For any even integer x, we have x + 0 = 0 + x = x, as the sum of any even integer with 0 is itself. Thus, the existence of an identity element holds for E under addition.
4. Existence of Inverse Elements (G4):
To prove the existence of inverse elements, we need to show that for every element x in E, there exists an element -x in E such that x + (-x) = (-x) + x = 0.
For any even integer x, its additive inverse is -x, another even integer.
To prove x + (-x) = 0, we have:
x + (-x) = 2a + (-2a) = 2(a - a) = 2(0) = 0.
Similarly, we can prove (-x) + x = 0 using the same steps.
Therefore, for every even integer x, there exists an additive inverse -x in E, satisfying the existence of inverse elements.
In conclusion, the set of all even integers E with the operation of addition forms a group because it satisfies all four group axioms: closure, associativity, existence of an identity element, and existence of inverse elements.