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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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?

Mathematics
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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
is that because they are uncountable?
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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well you will "see" two lines , y = 1 and y = 0
it doesnt matter there are more irrationals than rational, so what
Are there more irrational numbers than rational is really the question
as far as you are concerned, between any two irrationals is a rational, between any two rationals is an irrational, etc
or more generally, pick two points, between them is an irrational and a rational
pick 2 distinct points
so it is a scattering of individual points, neither line exists
very true, imagine we had a microscope ;)
right, its not a line ,. but at the macro level it "looks" like a line
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
that is a true statement, but that doesnt mean that you can order them , rational, irrational, rational, irrational, etc
on the number line. so ...
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 ?
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
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?
there cannot be two distinct irrationals without a rational between them, correct
so its kind of paradoxical
Do you have a good source where I could read more? Wiki gets too techie too quickly
yeah, thats true. well you can google cantor's infinity
but this question is not addressed specifically, its something that my math professor told me
there are many layman easy websites
Yeah, I want something between the two levels
yeah i know the feeling
Also, I think it's neat to see hebrew letters come into math for this question!
well college libraries have some gradual immersion books
yeah, thats cool
the big question in advanced math, does there exist a cardinality between aleph null ( natural numbers) and the cardinality of the real numbers
it can be shown that aleph null is the "least" infinity , as weird as that sounds
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
so we start with | N | , where N = { 1,2,3,...}
ok
where |N | means cardinality , that means the size of the set
right
now we know for any set , |P (S)| > |S|
P(S) is the power set of S
only for finite sets?
for any set
ok
it comes in handy for infinite sets
what a power set ?
2^S, right?
the number of permutations within the set
right, thats the shorthand for it
well, its the set of all subsets of the set
i dont know about permutations, hmmm
I thought infinity was defined as the set that has the same cardinality as any of its proper subsets
yes thats a definition of an infinite set
but im talking about power set, what is the power set operation
ok say you have { 1, 2 } , the power set is { {}, {1}, {2} , {1,2} }
S= { 1, 2 } , then P(S)= { {}, {1}, {2} , {1,2} }
right
now cantor proved a delightful theorem , that | P ( S) | > | S| for any set S
so the powerset of integers is larger I mean has a higher cardinality than the set of integers
exactly
ok so
| P ( N ) | > | N | , correct ?
I sense a trick coming up...
but, yes
http://en.wikipedia.org/wiki/Cardinality_of_the_continuum
ok it turns out that |R| which we use shorthand c , it turns out |R| = | P ( N ) | ,
now the question is, is there an infinity between |N| and | P ( N ) |
is the real numbers the smallest infinity after the natural numbers
Wouldn't rational and irrational be smaller than real?
we have a hierarchy of infinities, |N| < |P(N)| < |P (P(N))| < ...
oh boy, rational is the same cardinality as natural numbers
the irrationals are uncountable
So cantor's work doesn't cover uncountable sets?
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.
help

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