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Where in math is the subset partial ordering used to describe one set as larger than another? Diagonalization isn't showing that a number in set A isn't in set
by pcmonk 10y ago
Where in math is the subset partial ordering used to describe one set as larger than another?
Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A.
We're obviously not using the same definition of "size". I generally think in terms of cardinality, what are you thinking of?
- Retric 10y agoCardinality is the number of elements in a set. If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C} There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.
- pcmonk 10y agoThat'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.
- 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.
- stouset 10y agoThe cardinality of the set of prime numbers and of rationals are both aleph-null[1]. Same with the list of all integers, the list of all positive integers, and the list of all even integers. They all have the same cardinality, and that cardinality is aleph-null. But please, continue to argue with mathematical definitions that have been established for over a century. [1]: https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught