Recent comments in /f/Showerthoughts

LisaThorpe t1_j29vu09 wrote

Until you realize that thinking someone’s personality is superior to yours makes you insecure, and they dump you. Then you feel twice as crappy, but also have no incentive to change since you know you can’t win them back. And then your life devolves into mental and physical self hate.

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Haven_Stranger t1_j29vkgd wrote

Two classes? What two classes? OP specified "numbers between 0 and 1" and "[numbers] from 0 to infinity". The class involved (entailed, even) is "numbers on the number line". In other words, reals. That's one class. That's the basis of the comparison.

Both specified ranges are uncountably infinite.

Here's an easier comparison: There are as many numbers between 0 and 1 as there are between 1 and infinity. It's an easier comparison because now both sets are strictly between their bounds, the two sets don't overlap, and the bijection formula is simpler.

If A is a real between 0 and 1, and B is the matching real greater than 1, then the mapping is:

A -> 1 / B

Also

B -> 1 / A

That's it. For every real number larger than one, there exists exactly one matching number between zero and one, and vice-versa. No exceptions, no excuses, nothing left unaccounted.

The size of the two sets are exactly the same, even though the extents of the two sets are wildly different.

Also also, that's the comparison OP meant to express. Even so, the comparison posted still holds true. It just that, instead of mapping A to the inverse of B, we map A to the inverse of one more than B.

So, no, we don't have to compare reals on one side and rationals on the other, or anything else where we'd have to specify two classes. Those comparisons can be made, of course, but they're not relevant to the post.

It takes different real numbers to make it all the way out to infinity, but it doesn't take more of them. There are exactly as many real numbers greater than 1 as there are between 0 and 1.

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Arganancer t1_j29uh5s wrote

Yeah you're absolutely right. It becomes abundantly clear when you replace "thrive" with one of its synonyms like "prosper" that it doesn't really work either way. Thanks for bringing it up!

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farineziq t1_j29u4qj wrote

I think finding a one to one relation between elements of one set to elements of another set is a convincing way to know if they have the same amount of elements without counting them. Like if you have two heaps of rocks, you could make two lines of rocks where rocks are side by side and if both lines end together, both sets have the same amount of rocks.

Doing the same with infinite sets is useful because we can't count the elements, but might be able to find a function that associates each element of one set to one and only one element of the other. (Regarding the rocks example, every possible way to put them side by side is such a function.)

For example, there are as many positive integers as positive and negative integers because f(x) = (x % 2) * (-x / 2 - 1 / 2) + (1 - x % 2) * (x / 2) from 0 to ∞ gives 0, -1, 1, -2, 2, -3, 3, ...

Now regarding uncountable infinite sets like the real numbers, someone else in the comment showed that ]0, 1] has the same amount of elements as [1, ∞ [ using f(x) = 1 / x. And regarding what we're actually looking for, let's say [0, 1] has the same amount of elements as ℝ, I didn't really pay attention but this seems convincing.

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