Set Theory (Part 15): Dedekind Cut Arithmetic

Please feel free to leave comments/questions on the video and practice problems below!

In this video, we will set up arithmetic (addition and multiplication) for Dedekind cuts and thereby show that the real numbers form a complete ordered field. We will also prove some common laws of real number arithmetic, such as the commutative law for addition and multiplication and the cancellation law.

[robertwilsoniii]

Great examples at 4:10, 14:30 and17:50. For the definition of absolute value, if you get stuck, remember that x is a real number and so x is a set, a dedekind cut, so when we do this we look only at the lub of each cut, so if you wanted to look at -1, say, then you need the cuts -1 and 1 (which are both sets) so you have to make new sets every time you consider new values of x. Don't try to pick any rational values in some other cut that is closed downwards; you need new cuts, one for each real number x you want the absolute value of, and then it should become clear that you union these two sets and find your value. Okay, now if you think about this it's also pretty clear why |x| > 0 for all x in R. Happy learning!

Multiplication starts at 18:00.

[amyxst]

I'm wondering if there's a mistake for the slide at 7:55? At the very bottom, you say that: 2 is not an element of x => -2 is an element of -x. Should the implication not be: 2 is not an element of x => 2 is an element of -x?

[brianlonghuixu]

I am wondering how to prove the lemma at 12.16.

[laflaca]

hi, how should we define under this construction x^y, provided both x and y are real numbers and y is positive?

[arghyachakraborty]

At 28:45 What will be the solution of problem number 8 in practice problems??


2 weeks gone but there is no reply

[GianlucaUK]

Very useful video, but I wonder if you've ever actually tried to go through the proofs you mention as exercises in your last slide. Your bullet point number 8 is extremely painful. It is fairly straightforward to prove that x * 1/x is a subset of 1, but proving that 1 is a subset of x * 1/x is a completely different ballgame. I see many textbooks "cheating" and saying they leave it as an exercise. Do you know of any books where these proofs are given?