Group Theory

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.

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.

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?