# Group TheoryWatch

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#1
Ok so I understand how to prove a group, it must satisfy these criteria:

1) Closure - Any two elements in that group, under the binary operation listed, should produce another element in the same group

2) Associativity - Order doesn't matter, e.g : a*(b*c) = (a*b)*c

3) Identity - Any element combined with the identity under the binary operation should give you the same element. Identity is 0 for addition and 1 for multiplication. The identity must also be in the group

4) Inverse - Any element combined with its inverse, under the binary operation listed, should give you the Identity. The Inverse must also be in the group.

So the text book I have gives me simple examples, groups with like only 4 elements.

Let's suppose I get the question

Prove (Z,+) is a group.

Z = Set of real numbers

I know any integer added with another integer gives you an integer (It's common sense), but how do I prove it? I obviously can't try out every integer in existence.
0
1 year ago
#2
(Original post by Doctor1234)
So the text book I have gives me simple examples, groups with like only 4 elements.

Let's suppose I get the question

Prove (Z,+) is a group.

Z = Set of real numbers

I know any integer added with another integer gives you an integer (It's common sense), but how do I prove it? I obviously can't try out every integer in existence.
As part of group theory, there is no requirement to prove that the sum of two integers is an integer.

If you do mathematics at uni., then you may do a course on the foundations of mathematics, with first the natural numbers, and then the integers being constructed from a set theoretic base, along with the operations of addition and subtraction, and so on.
0
1 year ago
#3
(Original post by Doctor1234)
Ok so I understand how to prove a group, it must satisfy these criteria:

1) Closure - Any two elements in that group, under the binary operation listed, should produce another element in the same group

2) Associativity - Order doesn't matter, e.g : a*(b*c) = (a*b)*c

3) Identity - Any element combined with the identity under the binary operation should give you the same element. Identity is 0 for addition and 1 for multiplication. The identity must also be in the group

4) Inverse - Any element combined with its inverse, under the binary operation listed, should give you the Identity. The Inverse must also be in the group.

So the text book I have gives me simple examples, groups with like only 4 elements.

Let's suppose I get the question

Prove (Z,+) is a group.

Z = Set of real numbers

I know any integer added with another integer gives you an integer (It's common sense), but how do I prove it? I obviously can't try out every integer in existence.
You don’t really need to prove it explicitly as it’s obvious.
A proof for it may follow from Peano axioms but it’s just unecessary work. I don’t see a neat way to prove it explicitly.
0
#4
(Original post by ghostwalker)
As part of group theory, there is no requirement to prove that the sum of two integers is an integer.

If you do mathematics at uni., then you may do a course on the foundations of mathematics, with first the natural numbers, and then the integers being constructed from a set theoretic base, along with the operations of addition and subtraction, and so on.
So I can literally say

a,b,c E Z (a, b and c are in the set of integers)
: a+b=c?

Also, you said I don't need to prove 2 integers add to make another integer. What sets and what operations do I need to prove then?
0
1 year ago
#5
(Original post by Doctor1234)
So I can literally say

a,b,c E Z (a, b and c are in the set of integers)
: a+b=c?
Not sure what you mean by that. Can you put it into english.

I would simply state, "the sum of any two integers is an integer, and so Z is closed under addition."

Also, you said I don't need to prove 2 integers add to make another integer. What sets and what operations do I need to prove then?
Anthing that's not totally obvious.
0
1 year ago
#6
I would suggest googling ‘axioms of real numbers’. That will sort you out
0
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