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Topic: rational nubers and set roster (Read 6080 times) |
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minty33
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rational nubers and set roster
« on: Nov 16th, 2012, 1:44pm » |
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Its understood that rational numbers and real numbers can't be shown in set roster notation because they are infinitely uncountable. My discrete math teacher said there is actually a way to do it and challenged us to come up with it. I actually came up with a way but he said there is a way that doesn't involve sets inside the main set. Basically my way was:
{ ...,{(...,-3,-2,-1)-2/1(1,2,3,...)},{(...,-3,-2,-1)-1/1(1,2,3,...)},0,{( ...,-3,-2,-1)1/1(1,2,3,...)},{(...,-3,-2,-1)2/1(1,2,3,...)},...}
Note that the ordered pairs apply to the denominator. I just can't show that typing it like I did handwritten.
So how would I answer his challenge of writing rationals in set roster without using other sets. He said the hotel infinity riddle is a hint but I can't figure out how. I guess it is a simpler way of showing the incrimination but I don't get how it applies here. Any ideas of how else to show rationals in set roster notation.
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| « Last Edit: Nov 16th, 2012, 2:40pm by minty33 » |
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towr
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Re: rational nubers and set roster
« Reply #1 on: Nov 16th, 2012, 2:10pm » |
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Rationals are not infinitely uncountable. Reals are, but rationals are as countable as integers.
I'm not familiar with the terminology 'set roster notation' (not having followed any discrete mathematics courses in English). But one way to count all rationals (some multiple times, for simplicity), is for example, to take an integer and map the odd-position digits to a new integer A, the even-position digits to B, then take A/B. So for example, -1928476 => -1246/987 And if there's a mapping from a countably infinite set to a set S, then S is also countable.
And there's literally countless other ways. Another popular way is to spiral outward around 0,0 in the Cartesian grid (skipping divisions by 0 and doubles).
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pex
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Re: rational nubers and set roster
« Reply #2 on: Nov 16th, 2012, 2:18pm » |
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I also needed to google it, but apparently "set roster notation" is just the standard {a, b, c, ...} notation.
To list the positive rational numbers, one could for example first take those with numerator+denominator=1, then =2, =3, ... So:
{0/1, 0/2, 1/1, 0/3, 1/2, 2/1, 0/4, 1/3, 2/2, 3/1, 0/5, 1/4, 2/3, 3/2, 4/1, ...}
To list all rational numbers, just add the corresponding negative after each positive number.
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minty33
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Re: rational nubers and set roster
« Reply #3 on: Nov 16th, 2012, 2:39pm » |
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Thanks guys I sorta get it. I didn't realize rationals where countable. My notes are wrong. I'll ask my teacher Monday if i Copied something wrong. That said you are right. This chart helped. http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
But it doesn't seem to reflect either of your counts or its over my head at the moment. I'll reflect some more but thanx again. In roster notation though you have to show a repeatable pattern before you can use the eclipses(...) I think. That's where I'm falling short of fully getting your answers as I can't recognize the pattern but that's probably just me. By the way doesn't my way show all real numbers. It repeats them as well but that's allowable in sets. If I increment the denominator with integers as a subset of incrimenting the numerator as integers of the subset of the main set. Any way that's confusing to say in English but you can see my answer. Does that not represent all real numbers.
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pex
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Re: rational nubers and set roster
« Reply #4 on: Nov 16th, 2012, 2:56pm » |
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on Nov 16th, 2012, 2:39pm, minty33 wrote:Thanks guys I sorta get it. I didn't realize rationals where countable. My notes are wrong. I'll ask my teacher Monday if i Copied something wrong. That said you are right. This chart helped. http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
But it doesn't seem to reflect either of your counts or its over my head at the moment. I'll reflect some more but thanx again. |
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That's pretty much what I did, except they skip over zeros and other duplicates, and for some reason they change directions for every diagonal.
on Nov 16th, 2012, 2:39pm, minty33 wrote:| In roster notation though you have to show a repeatable pattern before you can use the eclipses(...) I think. That's where I'm falling short of fully getting your answers as I can't recognize the pattern but that's probably just me. |
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Maybe it's easier to recognize if I write it like this:
{0/1,
0/2, 1/1,
0/3, 1/2, 2/1,
0/4, 1/3, 2/2, 3/1,
0/5, 1/4, 2/3, 3/2, 4/1,
...}
on Nov 16th, 2012, 2:39pm, minty33 wrote:| By the way doesn't my way show all real numbers. It repeats them as well but that's allowable in sets. If I increment the denominator with integers as a subset of incrimenting the numerator as integers of the subset of the main set. Any way that's confusing to say in English but you can see my answer. Does that not represent all real numbers. |
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It definitely doesn't list all real numbers, as that set is uncountable. It might list all rational numbers, but I'm afraid I don't quite understand your notation.
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minty33
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Re: rational nubers and set roster
« Reply #5 on: Nov 16th, 2012, 3:36pm » |
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OK that is easier to understand. Now I see. From cantors argument I came up with
{1/1,1/2,1/3,...,2/1,2/2,2/3...3/1,3/2,3/3...,...}
Note that the last elipses shows the overall pattern repeats the elipses right before it shows the denominator inceimentation.
Problem is technically they are subset again which is right but not part two of his challenge. now I see you can do It the other way because I see your pattern now. as that chart does switch directions it made a pattern hard to show. Thank you.
I know it was hard to understand my original notation unless I showed it to you handwritten.
Basically inside each {} is an infinite countable set e.g {(...,-3,-2,-1) 1/1 (1,2,3...)} inside the ordered pair () is the incrimentation for the denominator of an integer represented in rational form, 1/1 in this case. So it shows {....,1/-3,1/-2,1/-1,1/1,1/2,1/3...}. That's one subset. the next subset {} is also inside the main set {} and is the same except for 2/1 is used not 1/1 and that shows infinite possibilities for 2/1. The subsets pattern is repeated. For each integer that is represented in 1/1, 2/1 form and all possible denominators for each integer represented that way is shown in each subset using integers inside(). What this means is it accounts for all decimals of every number but obviously there is crossover but duplicates are allowed in users so it doesn't matter. I'm not saying that reals are infinitely countable I'm just asking if this does show a representation for all of the in set roster. They are not countable in the sense you never get to two from one but I believe my set shows the set of all real numbers in set roster notation as opposed to the much more efficient and preferred set builder notation.
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| « Last Edit: Nov 16th, 2012, 3:44pm by minty33 » |
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peoplepower
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Re: rational nubers and set roster
« Reply #6 on: Nov 16th, 2012, 3:53pm » |
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I failed to understand what you wrote, but it sounds like you are taking a union, clearly not allowed in set roster notation.
It's not exactly trivial to do this in set builder notation either.
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pex
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Re: rational nubers and set roster
« Reply #7 on: Nov 16th, 2012, 3:57pm » |
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Thanks for explaining your notation. Yes, it lists all of the rational numbers. However, it is not roster notation, as that does not allow for infinite "sets within sets". You're basically saying
{..., {all rationals with numerator -2}, {all rationals with numerator -1}, 0, {all rationals with numerator 1}, {all rationals with numerator 2}, ...}
which clearly contains all rationals, but does not do much to show that that set is countable. (Unless you've already proven that a countable union of countable sets is countable, of course.)
It still does not contain each real number: where is sqrt(2), for example?
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minty33
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Re: rational nubers and set roster
« Reply #8 on: Nov 16th, 2012, 4:11pm » |
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I'm pretty sure its not a union. A union contains all the elements of two or more sets. I am just making a set of sets which is allowed from what my teacher has taught us and he did say my way is a way to show all rationals just not the clearest way. A Set in a set is e.g {{1,2,3,...},{...,-3,-2,-1}} and is allowed and nothing says these sets can't be infinite in their own right. Similarly I also can show {{1},{2},{3},...} and means this pattern of sets repeats. This is not equivalent to above but its these two ideas I combined. I used sets in the main set who's integer pattern repeat 1/1 2/1 ect.. but inside those sets ordered pairs show the denominator increment pattern that give it its infinite property. What you end up with is an infinite number of elements inside a set and a infinite number of sets of said sets who differ by the rational representation of integers 1/1,2/1 thus giving a different set of infinte numbers for each. Thus it is a set of infinite sets each containing infinite elements.
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| « Last Edit: Nov 16th, 2012, 4:18pm by minty33 » |
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minty33
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Re: rational nubers and set roster
« Reply #9 on: Nov 16th, 2012, 4:23pm » |
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Your right it doesn't show countability i was using that term wrong from the beginning but I thought I can use infinite sets. E.g {{1},{2},{3},...}. But maybe not you know better than me. I actually didn't show them as seperate sets when I showed my teacher so maybe I just don't use the curly braces. Wh knows. But thank you for all your info.
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pex
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Re: rational nubers and set roster
« Reply #10 on: Nov 16th, 2012, 4:25pm » |
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on Nov 16th, 2012, 4:11pm, minty33 wrote:| I'm pretty sure its not a union. A union contains all the elements of two or more sets. I am just making a set of sets which is allowed from what my teacher has taught us and he did say my way is a way to show all rationals just not the clearest way. A Set in a set is e.g {{1,2,3,...},{...,-3,-2,-1}} and is allowed and nothing says these sets can't be infinite in their own right. Similarly I also can show {{1},{2},{3},...} and means this pattern of sets repeats. This is not equivalent to above but its these two ideas I combined. I used sets in the main set who's integer pattern repeat 1/1 2/1 ect.. but inside those sets ordered pairs show the denominator increment pattern that give it its infinite property. What you end up with is an infinite number of elements inside a set and a infinite number of sets of said sets who differ by the rational representation of integers 1/1,2/1 thus giving a different set of infinte numbers for each. Thus it is a set of infinite sets each containing infinite elements. |
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Okay... but if you didn't intend it as a union, then the rational number 1/2 is not an element of your set! It is an element of "the set of all rationals with numerator 1", which is in turn an element of your large set. In that case, your set does not contain all rational numbers. In fact, it contains only one of them: the number 0.
This may seem like nit-picking, but if you're just starting to learn about set theory, it's probably good to be very (extremely) strict about notation. Just to remove one source of possible confusion 
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minty33
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Re: rational nubers and set roster
« Reply #11 on: Nov 16th, 2012, 4:43pm » |
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1/2 is a member of the set {(...,-3,-2,-1)1/1(1,2,3,...)} so is -1/2. Where it gets duplicated is in {(...,-3,-2,-1)2/1(1,2,3,...)} as 2/4 and again for each set e.g in 3/1's set its 3/6. I don't know really anything about set theory actually we skimmed it. Really this came up in class today I forget how. I think we were covering predicates and he used some sentence on rational numbers and asked if it was a statement. Then we got sidetracked on number systems. My class is on logic arguments( modus ponens, modus tolens, elimination ect...) , equivelences, truth tables and validity of arguments so far. Its discrete math 1. As for sets we just briefly covered some basic ideas the first week.
I didn't intend a union I just tried to represent all rationals somehow.
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| « Last Edit: Nov 16th, 2012, 4:47pm by minty33 » |
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minty33
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Re: rational nubers and set roster
« Reply #12 on: Nov 16th, 2012, 4:49pm » |
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Oh I see what you meant. Right it doesn't show it as an element. And 0 is not rational I don't think since 0/0 isn't allowed. I added that when we talked about reals. As I said originally I never had them as sets from a notation standpoint. I added the {} in an attempt to be more correct but that actually made it more wrong. I wonder if I left the braces out if it is better or can't I do that because each comma would be a range of numbers. Oh well. Thanks. I'm sure you got better things to do than teach me on a Friday night so I appreciate it.
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| « Last Edit: Nov 16th, 2012, 4:55pm by minty33 » |
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pex
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Re: rational nubers and set roster
« Reply #13 on: Nov 16th, 2012, 4:57pm » |
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on Nov 16th, 2012, 4:43pm, minty33 wrote:1/2 is a member of the set {(...,-3,-2,-1)1/1(1,2,3,...)} so is -1/2. Where it gets duplicated is in {(...,-3,-2,-1)2/1(1,2,3,...)} as 2/4 and again for each set e.g in 3/1's set its 3/6. I don't know really anything about set theory actually we skimmed it. Really this came up in class today I forget how. I think we were covering predicates and he used some sentence on rational numbers and asked if it was a statement. Then we got sidetracked on number systems. My class is on logic arguments( modus ponens, modus tolens, elimination ect...) , equivelences, truth tables and validity of arguments so far. Its discrete math 1. As for sets we just briefly covered some basic ideas the first week.
I didn't intend a union I just tried to represent all rationals somehow. |
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Oh, yes, 1/2 is definitely a member of
{..., 1/-2, 1/-1, 1/1, 1/2, ...}, and of
{..., 2/-2, 2/-1, 2/1, 2/2, ...}, and of
{..., 3/-2, 3/-1, 3/1, 3/2, ...}, etc.
However, 1/2 is not a member of {...,
{..., 1/-2, 1/-1, 1/1, 1/2, ...},
{..., 2/-2, 2/-1, 2/1, 2/2, ...},
{..., 3/-2, 3/-1, 3/1, 3/2, ...}, ...},
which is the set you're describing.
On the other hand, 1/2 is a member of the union of
{..., 1/-2, 1/-1, 1/1, 1/2, ...} and
{..., 2/-2, 2/-1, 2/1, 2/2, ...} and
{..., 3/-2, 3/-1, 3/1, 3/2, ...} etc.
This sort of subtle differences can just turn out to be very important to keep in mind. Not just in set theory, but in most of mathematics.
on Nov 16th, 2012, 4:49pm, minty33 wrote:| Oh I see what you meant. Right it doesn't show it as an element. And 0 is not rational I don't think since 0/0 isn't allowed. I added that when we talked about reals. As I said originally I never had them as sets from a notation standpoint. I added the {} in an attempt to be more correct but that actually made it more wrong. I wonder if I left the braces out if it is better or can't I do that because each comma would be a range of numbers. Oh well. Thanks. I'm sure you got better things to do than teach me on a Friday night so I appreciate it. |
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Don't worry, you're welcome!
One thing though (before I go spend the rest of Friday night on other things): zero is definitely a rational number! 0/0 is indeed not allowed, but 0/1 is. (And 0/2, 0/-1, ...)
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| « Last Edit: Nov 16th, 2012, 5:00pm by pex » |
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