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That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very
by pcmonk 10y ago
That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.
There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.
Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets https://en.wikipedia.org/wiki/Cardinality#Infinite_sets
- Retric 10y agoWe agree that {A,B,C} has a lower cardinality than {A,B}. Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.
- yongjik 10y ago1. e + ei is not real. 2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise. This function maps all real numbers to irrational numbers, 1-to-1.
- Retric 10y agoe is real, e + ei is irrational.
- pcmonk 10y agoI believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.
- Retric 10y agoFor every point on your mapimg I define two points X + 1 and x + 1i. You can assign infinity to one of them but not both.
- yongjik 10y agoI'm afraid you have some basic confusion with mathematical concepts, including "mapping", "irrational", "complex", and "infinity". So, after you define x+1 and x+i, now what? Also please bear in mind that "infinity" is neither real nor complex: if you have a well-defined mapping into complex numbers, then by definition, it never maps to infinity. (Yes, there are some "functions" like y = 1/x that "maps to infinity", but it's simply mathematicians being lazy and abusing notations because everybody around them understands what's going on.)
- Retric 10y agoO, I get it. There is a basic contradiction in set theory. Called Russell's paradox, there are two ways around it. First is ignoring it, aka everything builds from it's self nothing can become recursive. Or Zer's something or other that basically removed membership and equality and hides in the corner crying. As such infinity is generally assumed not to exit in R. And causes all this all infinite set's map able to each other are equivalent size crap. It's also why real mathematicians laugh at the set guys. But, sorry the way R was initially defined it included infinity and you only get to put it into 1 place on your mapping. Or as a math professor said, what angle is the highest number in R mapping to.
- yongjik 10y agoWhere did you get all this idea? You know enough terms, yet somehow you have an incorrect understanding of pretty much everything. If you are interested, please read an actual math textbook. (Yes, they can be a giant time sink, but at least you'll learn the correct meanings of sets and functions.)
- Retric 10y agoI have taken plenty of high level math classes for fun and easy A's, I even had a department head yell at me for not getting a PHD. So, I can speak the lingo. But, the absurd results are not generalizable outside of their assumptions sorry Axioms. Set theory being one of the most obvious cases. It's sadly like a religion in many ways, follow enough false statements and you can prove anything. Yet, if you find a contradiction then don't actually accept at least one of your assumptions are false.
- jakub_h 10y agoe is irrational, e+ei can't be irrational because it isn't real (all irrationals are reals).
- Retric 10y agoSorry, imaginary number composed of two irrational numbers.
- pcmonk 10y ago{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false. For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-simple-constructive-1-1-mapping-between-the-reals-and-the-irrationa http://math.stackexchange.com/questions/512397/is-there-a-si...
- Retric 10y agoThe distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.
- pcmonk 10y ago> The distance between 0 and 1 is smaller than the distance between 0 and 2. This is correct. However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2. > The number of points between 0 and 1 is larger than the number of rational numbers. This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.