References:
Copyright: JSTOR and it does not give a specific type of copyright.
http://students.mimuw.edu.pl/~tt249057/other/ksiazki%20i%20papery/talagrand-supremum%20of%20some%20canonical%20processes.pdf
This resource provides great insight for supremums. It also gives many examples along with their answers at the end of the online text.
Copyright: Wolfram Research and this site also does not say what type it just says that it is copyrighted and the years as well.
http://mathworld.wolfram.com/FieldAxioms.html
This site gives information on field axioms which is helpful in thinking about them. It shows some key properties of field axioms broken down easier for us to follow.
Exercises: Page 8
1.3.5
1.4.2
1.6.3
1.6.21
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In section 1X, we learn about many axioms that play a vital role in our Mathematics world. These are properties that apply to our computations and also back up our reasoning in proofs.
https://faculty.math.illinois.edu/~hildebr/347.summer14/completeness.pdf
*This link provides definitions and brief explanations for the axiom of Completeness, the Archimedean principle as well as other definitions that are provided in the next section. It comes from A.J Hildebrand but it doesn’t say the copyright.
The Axiom of Completeness states every set E of real numbers that is non-empty and bounded above has a supremum supE that is a real number.
The Archimedean Principle says, the set of natural numbers N has no upper bound. In other words, given any real number x, there exits a natural number n, such that n > x.
In this video, Dr. Salomone explains the Archimedean principle and shows us how to use it in a proof (minute 1:11-end).
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https://via.hypothes.is/https://doc-0g-50-docs.googleusercontent.com/docs/securesc/ha0ro937gcuc7l7deffksulhg5h7mbp1/154lr7pb7cmhbag2a8vm26usiodgr9dc/1513166400000/02480651323841671196/*/0B8-xKlYA8qaPYjdGbi1qYjItZFk?e=download
*This link is to our book which will provided us with the definitions we need.
In this section is also the Order Structure. The Order structure consists of four axioms that use inequalities to describe the reals and how they relate to each other along with operations used with them. This screenshot from our book shows us the four axioms of the Order Structure:
These facts are essential when writing proofs. Just like the Field Axioms, these are properties that may seem like mere facts, but they are useful for proofs. We want to leave no room for question, and this helps us do it.
https://via.hypothes.is/https://doc-0g-50-docs.googleusercontent.com/docs/securesc/ha0ro937gcuc7l7deffksulhg5h7mbp1/154lr7pb7cmhbag2a8vm26usiodgr9dc/1513166400000/02480651323841671196/*/0B8-xKlYA8qaPYjdGbi1qYjItZFk?e=download
*This link is to our book which will provided us with the definitions we need.
In our book, pages 6 and 7 list the nine Field Axioms that we are accustomed to using in early mathematics. Although they may not be introduced as “Axioms”, in the elementary level these are properties that apply to addition and multiplication. Here is a video that provides additional insight:
One of the packets we worked on has the following exercise:
In this exercise, we can practice using the learned Field Axioms to prove something. We must prove that the statement where (x+1)^2 equals x^2+2x+1 is true. On one hand, we could simply explain and show that foiling (x+1)^2 would result with the presented polynomial; but proving and supporting it with the axioms makes our proof more solid. Go ahead and try it for yourself!