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Nov
25
revised irrational numbers
added 3 characters in body
Nov
25
answered irrational numbers
Nov
25
comment How many distinct unlabeled binary trees with i leaves?
@i3wangyi: You’re very welcome.
Nov
25
answered Double complement law proof
Nov
25
revised Is it possible to flip tails indefinitely?
Spelling.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@user99680: You said that under $\mathsf{CH}$ they do not have the same cardinality; mentioning $\mathsf{CH}$ in this connections makes no sense unless you thought that in the absence of $\mathsf{CH}$ they might have the same cardinality. I’m simply pointing out that they don’t, ever, so that the mention of $\mathsf{CH}$ is completely irrelevant.
Nov
25
answered How many distinct unlabeled binary trees with i leaves?
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@user99680: But that’s simply not what uncountable means, ever, and $\mathsf{CH}$ has nothing to do with whether $\Bbb R$ and $\wp(\Bbb R)$ have the same cardinality: they don’t, full stop. See Cantor’s theorem.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@Andreas: I think so, though the description is awkward enough that I’m not quite sure. See if this is what you have in mind. For each $x\in M$ let $M_x=\{x\}\times M$; then $\{M_x:x\in M\}$ is a partition of $M\times M$ into uncountably many uncountable sets — specifically, into $|M|$ sets each of cardinality $|M|$. Then if $h:M\times M\to M$ is a bijection, $\{h[M_x]:x\in M\}$ is a partition of $M$ into uncountably many uncountable sets.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@user99680: $|\wp(\Bbb R)|$ is never equal to $|\Bbb R|$; this has nothing to do with CH or GCH.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@user99680: No, absolutely not. $\wp(\Bbb R)$ is an uncountable set admitting no bijection with $\Bbb R$. Uncountable simply means not countable, i.e., not finite and not countably infinite.
Nov
25
comment Need help with a combinations question
@FrostyStraw: No, four positions will be chosen from the set of ten positions. We don’t really care about the possible characters. It would be exactly the same calculation if we wanted to know how many different sets of four digits there are, since the set of all digits is a ten-element set: in both problems we’re just counting the number of ways of picking $4$ things from a set of $10$ things. It doesn’t matter whether the things are positions in a string, digits, people (if we’re forming committees of $4$ from a pool of $10$ people), or what.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
@user99680: What you’ve done works for $\Bbb R$ and is the application to $\Bbb R$ of the idea that I had in mind, but your attempt to extend it to arbitrary uncountable $S$ doesn’t work.
Nov
25
comment Existence of uncountable set of uncountable disjoint subsets of uncountable set
If $S$ is an arbitrary uncountable set, there need not be a bijection between $\Bbb R^2$ and $S$.
Nov
25
comment Need help with a combinations question
@FrostyStraw: $10\cdot9\cdot8\cdot7$ should not be divided by $(10-4)!$: it already is $\frac{10!}{6!}$. The division by $4!$ is correct, however, and leaves you with $\frac{10!}{6!4!}=\binom{10}4$. You can think of it that way, but in my opinion it’s an unnecessarily complicated way of looking at it. It’s much easier just to think in terms of picking subsets of a fixed size, which are counted by binomial coefficients.
Nov
25
answered Existence of uncountable set of uncountable disjoint subsets of uncountable set
Nov
25
comment Does current foundation of first order logic need a fundamental change?
@SaintGeorg: Nothing is wrong here. All you’re saying is that in first order logic the word theory has a precise meaning that does not include all of the connotations of the everyday sense(s) of the word. That’s not at all unusual when everyday words are given precise technical meanings in some discipline.
Nov
25
comment Does current foundation of first order logic need a fundamental change?
Most natural languages have never been written. The natural form of every natural language is either speech or, in the case of Nicaraguan Sign Language, gesture. It isn’t even true that the atomic elements of natural languages are words, since there is no satisfactory cross-linguistic definition of word. You might be able to justify a rough equivalence between mathematical symbols and morphemes, though even that is pretty shaky. That said, I agree with Trevor that proofs are a better analogue of texts.
Nov
25
answered Are there any relations R of size 15 on the set {1, 2, 3, 4, 5, 6} such that R is both transitive and symmetric?
Nov
25
answered Which strings belong to the regular set represented by the regular expression (1∗01∗0)?