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.
I would assume you would carry out the process of multiplying dedekind cuts. You would multiply x by itself y times, as this is the purpose of exponentiation. You might have to get clever with decimal expansions and go term by term.
You actually need Analysis to define exponentials. Basic powers for n in Z is easy, repeated multiplication. Define the series E(x)=Σ(x^n)/n!. Show all beautiful properties such as: E(x) converges for all x in R, it is biyective from R onto R+, E(x+y)=E(x)E(y) and E(n)=E(1)^n. Call log(x) the inverse. Now define x^y= E(ylog(x)).
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?
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?
+GianlucaUK At the time of making the videos and posing the questions I do actually make sure I can prove these statements. I haven't thought of this particular proof in a while, but it very well may be painful. My main reference is Enderton's _Elements of Set Theory_ and you might be able to find it in there. If not, it might be on ProofWiki or somewhere else online.
I thought as you it was painful but it's not that painful if you approach it correctly it can be proved as mathematical argument of infinite sequence but using algebraic methods to prove it was dead end for me !
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.
I am wondering how to prove the lemma at 12.16.
At 28:45 What will be the solution of problem number 8 in practice problems??
2 weeks gone but there is no reply
hi, how should we define under this construction x^y, provided both x and y are real numbers and y is positive?
I would assume you would carry out the process of multiplying dedekind cuts. You would multiply x by itself y times, as this is the purpose of exponentiation. You might have to get clever with decimal expansions and go term by term.
You actually need Analysis to define exponentials. Basic powers for n in Z is easy, repeated multiplication. Define the series E(x)=Σ(x^n)/n!. Show all beautiful properties such as: E(x) converges for all x in R, it is biyective from R onto R+, E(x+y)=E(x)E(y) and E(n)=E(1)^n. Call log(x) the inverse. Now define x^y= E(ylog(x)).
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?
So helpful :) Thanks.
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?
+GianlucaUK
At the time of making the videos and posing the questions I do actually make sure I can prove these statements. I haven't thought of this particular proof in a while, but it very well may be painful. My main reference is Enderton's _Elements of Set Theory_ and you might be able to find it in there. If not, it might be on ProofWiki or somewhere else online.
Thanks!
I thought as you it was painful but it's not that painful if you approach it correctly it can be proved as mathematical argument of infinite sequence but using algebraic methods to prove it was dead end for me !
@GianlucaUK hey if you know how to prove 1* is subset of x×1/x then please help me to do it...