Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
An introduction to number systems, ordered sets, and their properties. It covers the concept of sets, ordered sets, and their definitions, examples of ordered sets, and the existence of least upper and greatest lower bounds. The document also introduces the concept of fields and their axioms, examples of fields, and theorems related to fields in an ordered context.
Typology: Study notes
1 / 38
This page cannot be seen from the preview
Don't miss anything!
Ordered SetsFields
The most first numbers every considered were the whole numbers:
1 , 2 , 3 ,...
Ordered SetsFields
The most first numbers every considered were the whole numbers:
1 , 2 , 3 ,...
Then someone realized that it was important to include a number representing “nothing”. This then gave us the natural numbers:
N = 1, 2 , 3 ,...
Ordered SetsFields
Then people noticed that addition worked better if there were negative numbers. This led us to the integers
Z =... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,...
After dealing with the integers for a while people began to notice the usefulness of fractions and the rational numbers were born:
Q =
{ (^) a b : a, b ∈ Z
Ordered SetsFields
What is next...?
Ordered SetsFields
Definition A set is a collection of objects. If a set has at least one element we say it is non-empty. If a set has no objects we say it is empty.
Definition Suppose A is a set If x is a member of A we write x ∈ A. If x is not a member of A we write x 6 ∈ A.
Ordered SetsFields
Definition Suppose A and B are sets If every element of A is an element of B we say A is a subset of B and write A ⊆ B or B ⊇ A If A is a subset of B and not equal to B we say A is a proper subset of B If A ⊆ B and B ⊆ A then we say the sets are equal and write A = B.
Ordered SetsFields
Definition An ordered set is a set S in which an order is defined.
Definition If S is an ordered set with < and x < y , we often say x is less than y. WE also often use y > x in place of x < y when convenient.
We will use x ≤ y as a shorthand for x < y or x = y. i.e. x ≤ y if and only if (NOT y < x)
Ordered SetsFields
Here are some examples The one point set {∗} with nothing satisfying <
Ordered SetsFields
Here are some examples The one point set {∗} with nothing satisfying < Z with the order a < b if and only if b − a is positive. Z with the order a < b if either |a| < |b| |a| = |b|, a is negative and b is positive. Notice that if S is an ordered set with < and E ⊆ S then E is an ordered set with <.
Ordered SetsFields
Definition Suppose S is an ordered set and E ⊆ S. If there exists β ∈ S such that x ≤ β for all x ∈ E then we say E is bounded above and call β an upper bound. If there exists β ∈ S such that x ≥ β for all x ∈ E then we say E is bounded below and call β an lower bound.
Ordered SetsFields
Lets consider the set Q with the standard ordering. Let X = {q ∈ Q : q ≥ 0 and q ≤ 1 }
Ordered SetsFields
Lets consider the set Q with the standard ordering. Let X = {q ∈ Q : q ≥ 0 and q ≤ 1 } X has a greatest lower bound and a least upper bound in X
Ordered SetsFields
Lets consider the set Q with the standard ordering. Let X = {q ∈ Q : q ≥ 0 and q ≤ 1 } X has a greatest lower bound and a least upper bound in X Let X = {q ∈ Q : q > 0 and q < 1 } X has a greatest lower bound and a least upper bound in Q but not in X.
Ordered SetsFields
Lets consider the set Q with the standard ordering. Let X = {q ∈ Q : q ≥ 0 and q ≤ 1 } X has a greatest lower bound and a least upper bound in X Let X = {q ∈ Q : q > 0 and q < 1 } X has a greatest lower bound and a least upper bound in Q but not in X. Let X = {q ∈ Q : q ≥ 0 }
