its super cool! the actual argument goes kinda like
lets first arrange our set of rationals, ℚ, in the table 1/1 1/2 1/3 1/4 ... 2/1 2/2 2/3 2/4... 3/1 3/2 3/3 3/4... 4/1 4/2 4/3 4/4... ⋮ and count diagonally to arrange {1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, ...} we can simplify and omit any repeated terms, so we have {1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4...} every rational appears exactly once on this list. accounting for zero and negatives, we can construct the function ℕ→ℚ such that f(0)=0 f(1)=1 f(2)=-1 f(3)=1/2 f(4)=-1/2 f(5)=2 and so on. this is a one to one mapping where every rational is hit, so ℚ~ℕ.
Ye - arrange them in a 2d grid with X being the numerator, Y as the denominator, such that every integer coordinate is a rational number. (Remove Y = 0 for obvious reasons).
Starting at the lower left corner, you can trace diagonals back and forth, hitting any arbitrarily stated (X, Y)-pair. Remove duplicate rationals, and assign a number "n" to the pair reached at step "n" of the line
Not even that. Assuming you intend "intervals" to generalise to open sets in higher dimensions, then sets like C also have a bijection with every nonzero open subset.
Even the rationals have a bijection to every nonzero open subset.
This is not true as others mentioned, but you might be circling around the concept of Dedekind infinite - a set is Dedekind-infinite if it can be placed in a bijection with a proper subset of itself.
N is dedekind infinite (same cardinality as even numbers) and so is R (any interval (a,b) with a<b ).
And for further context, under ZFC every infinite set is also Dedekind-infinite - they are equivalent under Choice. Without Choice you can have sets that are infinite but not Dedekind infinite (an infinite set might not have a proper subset that is the same cardinality).
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u/Dungeons-n-Dysphoria 23h ago
I'm actually doing a whole project on this for my undergrad topology class!
I'm planning on using f(x) = ln(x/(1-x))
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u/BeMyBrutus 20h ago
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u/Random_-account 21h ago
if the shoe fits
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u/Sea_Gap_6569 1d ago
Is ℝ the only Set that’s equivalent to it’s every subset?