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

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- dumbcow

i 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

is that because they are uncountable?

- dumbcow

kind of , rational numbers are uncountable too but for every gap between 2 rational numbers there could be an infinite number of irrational numbers

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- anonymous

well you will "see" two lines , y = 1 and y = 0

- anonymous

it doesnt matter there are more irrationals than rational, so what

- anonymous

Are there more irrational numbers than rational is really the question

- anonymous

as far as you are concerned, between any two irrationals is a rational, between any two rationals is an irrational, etc

- anonymous

or more generally, pick two points, between them is an irrational and a rational

- anonymous

pick 2 distinct points

- anonymous

so it is a scattering of individual points, neither line exists

- dumbcow

very true, imagine we had a microscope ;)

- anonymous

right, its not a line ,. but at the macro level it "looks" like a line

- anonymous

So 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

that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc

- anonymous

on the number line. so ...

- anonymous

it 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

between 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

okay, 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

there cannot be two distinct irrationals without a rational between them, correct

- anonymous

so its kind of paradoxical

- anonymous

Do you have a good source where I could read more? Wiki gets too techie too quickly

- anonymous

yeah, thats true. well you can google cantor's infinity

- anonymous

but this question is not addressed specifically, its something that my math professor told me

- anonymous

there are many layman easy websites

- anonymous

Yeah, I want something between the two levels

- anonymous

yeah i know the feeling

- anonymous

Also, I think it's neat to see hebrew letters come into math for this question!

- anonymous

well college libraries have some gradual immersion books

- anonymous

yeah, thats cool

- anonymous

the big question in advanced math, does there exist a cardinality between aleph null ( natural numbers) and the cardinality of the real numbers

- anonymous

it can be shown that aleph null is the "least" infinity , as weird as that sounds

- anonymous

and 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

so we start with | N | , where N = { 1,2,3,...}

- anonymous

ok

- anonymous

where |N | means cardinality , that means the size of the set

- anonymous

right

- anonymous

now we know for any set , |P (S)| > |S|

- anonymous

P(S) is the power set of S

- anonymous

only for finite sets?

- anonymous

for any set

- anonymous

ok

- anonymous

it comes in handy for infinite sets

- anonymous

what a power set ?

- anonymous

2^S, right?

- anonymous

the number of permutations within the set

- anonymous

right, thats the shorthand for it

- anonymous

well, its the set of all subsets of the set

- anonymous

i dont know about permutations, hmmm

- anonymous

I thought infinity was defined as the set that has the same cardinality as any of its proper subsets

- anonymous

yes thats a definition of an infinite set

- anonymous

but im talking about power set, what is the power set operation

- anonymous

ok say you have { 1, 2 } , the power set is
{ {}, {1}, {2} , {1,2} }

- anonymous

S= { 1, 2 } , then P(S)= { {}, {1}, {2} , {1,2} }

- anonymous

right

- anonymous

now cantor proved a delightful theorem , that
| P ( S) | > | S| for any set S

- anonymous

so the powerset of integers is larger I mean has a higher cardinality than the set of integers

- anonymous

exactly

- anonymous

ok so

- anonymous

| P ( N ) | > | N | , correct ?

- anonymous

I sense a trick coming up...

- anonymous

but, yes

- anonymous

http://en.wikipedia.org/wiki/Cardinality_of_the_continuum

- anonymous

ok it turns out that |R| which we use shorthand c , it turns out
|R| = | P ( N ) | ,

- anonymous

now the question is, is there an infinity between |N| and | P ( N ) |

- anonymous

is the real numbers the smallest infinity after the natural numbers

- anonymous

Wouldn't rational and irrational be smaller than real?

- anonymous

we have a hierarchy of infinities,
|N| < |P(N)| < |P (P(N))| < ...

- anonymous

oh boy, rational is the same cardinality as natural numbers

- anonymous

the irrationals are uncountable

- anonymous

So cantor's work doesn't cover uncountable sets?

- anonymous

I 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.

- anonymous

help

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