Formerly u/CanadaPlus101 on Reddit.

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Cake day: June 12th, 2023

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  • Yeah, that was actually an awkward wording, sorry. What I meant is that given a non-continuous map from the natural numbers to the reals (or any other two sets with infinite but non-matching cardinality), there’s a way to prove it’s not bijective - often the diagonal argument.

    For anyone reading and curious, you take advantage of the fact you can choose an independent modification to the output value of the mapping for each input value. In this case, a common choice is the nth decimal digit of the real number corresponding to the input natural number n. By choosing the unused value for each digit - that is, making a new number that’s different from all the used numbers in that one place, at least - you construct a value that must be unused in the set of possible outputs, which is a contradiction (bijective means it’s a one-to-one pairing between the two ends).

    Actually, you can go even stronger, and do this for surjective functions. All bijective maps are surjective functions, but surjective functions are allowed to map two or more inputs to the same output as long as every input and output is still used. At that point, you literally just define “A is a smaller set than B” as meaning that you can’t surject A into B. It’s a definition that works for all finite quantities, so why not?












  • CanadaPlus@lemmy.sdf.orgtoProgrammer Humor@lemmy.mlDesigners cry quietly
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    6 days ago

    Uhh, so looking carefully at the picture, it appears they shouldn’t have bothered with the inner pathway at all, and should have just connected the bridge over the canal (?) in the background to whatever is under the camera.

    Not only does the current design fail to provide a short path in demand, it leaves a goofy little boulevard behind the benches in what appears to be a dense, desirable urban area where you shouldn’t waste space.