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anonymous
 5 years ago
A piecewise function is defined as follows: for all rational numbers, f(x) = 1, for all irrational numbers, f(x) = 0. will the resulting graph be a line on f(x) = 0 with discontinuities? a line on f(x) = 1 with discontinuities, or a scattering of individual points?
anonymous
 5 years ago
A piecewise function is defined as follows: for all rational numbers, f(x) = 1, for all irrational numbers, f(x) = 0. will the resulting graph be a line on f(x) = 0 with discontinuities? a line on f(x) = 1 with discontinuities, or a scattering of individual points?

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i tend to agree with line on y=0 with discontinuities, there are far more irrational numbers than rational when looking at all real numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0is that because they are uncountable?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0kind of , rational numbers are uncountable too but for every gap between 2 rational numbers there could be an infinite number of irrational numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well you will "see" two lines , y = 1 and y = 0

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0it doesnt matter there are more irrationals than rational, so what

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Are there more irrational numbers than rational is really the question

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0as far as you are concerned, between any two irrationals is a rational, between any two rationals is an irrational, etc

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0or more generally, pick two points, between them is an irrational and a rational

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0pick 2 distinct points

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so it is a scattering of individual points, neither line exists

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0very true, imagine we had a microscope ;)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0right, its not a line ,. but at the macro level it "looks" like a line

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So the argument I have heard that because irrational numbers are uncountable, it is a "larger" infinity than the countable set of rational numbers, is false  or that there is only one more

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0on the number line. so ...

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0it gets even weirder. between any 2 rationals is an irrational. between any two irrationals is a rational. so where exactly are all the "extra" irrationals ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0between any 2 rational numbers there exists an irrational number. between any 2 irrational numbers there exists a rational number. so you might think, the number of rational and irrational should be about the same, since you can keep going with this statement

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0okay, you said, "that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc" does that mean that there can and cannot be two irrational numbers without a rational between them?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0there cannot be two distinct irrationals without a rational between them, correct

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so its kind of paradoxical

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Do you have a good source where I could read more? Wiki gets too techie too quickly

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yeah, thats true. well you can google cantor's infinity

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but this question is not addressed specifically, its something that my math professor told me

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0there are many layman easy websites

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yeah, I want something between the two levels

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yeah i know the feeling

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Also, I think it's neat to see hebrew letters come into math for this question!

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well college libraries have some gradual immersion books

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the big question in advanced math, does there exist a cardinality between aleph null ( natural numbers) and the cardinality of the real numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0it can be shown that aleph null is the "least" infinity , as weird as that sounds

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0and we can make a hierarchy of infinite cardinalities. use the natural number cardinality, and take the cardinality of the power set of the natural numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so we start with  N  , where N = { 1,2,3,...}

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0where N  means cardinality , that means the size of the set

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0now we know for any set , P (S) > S

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0P(S) is the power set of S

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0only for finite sets?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0it comes in handy for infinite sets

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the number of permutations within the set

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0right, thats the shorthand for it

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0well, its the set of all subsets of the set

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i dont know about permutations, hmmm

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I thought infinity was defined as the set that has the same cardinality as any of its proper subsets

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yes thats a definition of an infinite set

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but im talking about power set, what is the power set operation

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ok say you have { 1, 2 } , the power set is { {}, {1}, {2} , {1,2} }

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0S= { 1, 2 } , then P(S)= { {}, {1}, {2} , {1,2} }

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0now cantor proved a delightful theorem , that  P ( S)  >  S for any set S

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so the powerset of integers is larger I mean has a higher cardinality than the set of integers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0 P ( N )  >  N  , correct ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I sense a trick coming up...

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ok it turns out that R which we use shorthand c , it turns out R =  P ( N )  ,

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0now the question is, is there an infinity between N and  P ( N ) 

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0is the real numbers the smallest infinity after the natural numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Wouldn't rational and irrational be smaller than real?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0we have a hierarchy of infinities, N < P(N) < P (P(N)) < ...

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh boy, rational is the same cardinality as natural numbers

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the irrationals are uncountable

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So cantor's work doesn't cover uncountable sets?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I find it hard to assign the same cardinality to rationals as naturals, which means I only have a working understanding of cardinality and not a real grasp on it.
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