Integers

#matematik-discmath

We don't have subtraction yet, but know what we are aiming for; we know that once we have subtraction we can represent any integer as where . However, there are many ways express the same integer this way, so it makes sense to group together all possible ways to express the same integer into Equivalence classes. We know that , which is a relation solely on natural numbers! We also see that the relation is reflexive, , symmetric, , and transitive, so it is indeed an equivalence relation, meaning equivalence classes make sense.

Definition: Construction of the integers

Let , and define an equivalence relation on by The set of integers is the set of equivalence classes of with respect to . We define addition and multiplication on integers as follows: We also define and for every . For every we denote with .

Theorem

and for every .

Proof:
We have , so for every . We also have , so for every .

Theorem

If for some where , then .

Proof:
We have . Since , we have either or for some . If , then meaning giving . The proof is analogous for if .

Definition: Less than, greater than

Let . We say that is less than , denoted , if where and . If is less than we say that is less than and write .

Theorem

If such that and such that , then .

Proof:
We have where . This means We thus want to show that which is true because we know and .

Definition: Lower bound, upper bound

For a subset , an integer is a lower bound if for every , and an integer is an upper bound if for every .

Notice that since and do not need to be in , they are not unique.

Theorem

If a nonempty set has a lower bound, it has a least element.

Proof:
Let be a lower bound for , and let Thus for every we have , so meaning has a least element . We can then simply let to get , and we then have that for every , so , so is a least element of .