Brian M. Scott
Reputation
398/400 score
 Jan 31 answered Possible distinct binary tree structures at depth d Jan 31 revised Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ added 795 characters in body Jan 31 answered First axiom of countability and finer topologies Jan 31 answered Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$ Jan 31 answered Prime numbers making constant : 1.2527 Jan 31 comment Every tree has two leaves. Is my proof ok? This is excellent. Jan 31 comment Given an encryption key in a transposition cipher, find the decryption key I would need more information than is present in the actual problem statement. I can only guess that this information is implicit in the context of the material that you've been studying, to which I don't have access. Jan 31 comment Bachmann's construction of the real numbers @Ittay: It's fairly easy reading, though it does use some obsolete spellings and is a bit wordy by modern standards. The details turn out to be much like those for the Cauchy sequence approach. Jan 31 comment Propositional Logic Help $\neg p\land q$ can have either truth value; it need not be $F$. Do you have $p\to q\equiv \neg p\lor q$? Jan 30 comment Given an encryption key in a transposition cipher, find the decryption key What kind of transposition cipher? That key suggests columnar transposition, but for that you wouldn't have a separate decryption key. Jan 30 answered a bit complicated boolean simplification Jan 30 answered Different definitions of subnet Jan 30 comment Product of a First Countable Space by a Fréchet Space @Vinicius: I proved the only hard part starting with ‘If $q\in\operatorname{cl}_XA$’; what part is giving you trouble? Jan 30 comment Modify the Cantor pairing function @Nat: You don’t need to invert it to show that it’s bijective. Note that $$\frac{(m+1)m}2-\frac{m(m-1)}2=m\;,$$ so $\varphi(\langle 1,m+1\rangle)=\varphi(\langle m,m\rangle)+1$: there’s no overlap between the integers $\varphi(\langle k,m\rangle)$ with $1\le k\le m$ and the integers $\varphi(\langle k,m+1\rangle)$ with $1\le k\le m+1$. Jan 30 comment Monotone Convergence Theorem (for real sequences) equivalent to the Least Upper Bound Property? @user170039: I don’t agree that it’s a variant of the NIP. It’s actually a proof that the MCT implies the NIP, and at the same time it proves that the MCT implies the LUB property. Jan 30 comment Erdős-Szekeres theorem on monotone sequences You do indeed. You’re welcome! Jan 30 comment Is this strengthening of paracompactness known? @Henno: Or John Greever, ‘On Some Generalized Compactness Properties’. He was working on this stuff when I was taking his classes. Jan 30 revised Is this strengthening of paracompactness known? added 163 characters in body Jan 30 comment Is this strengthening of paracompactness known? @Henno: It's the same property. I learned it as hypocompactness as an undergraduate in the late $1960$s. I was pretty sure that it had a different name nowadays, but I couldn't remember what it was. Jan 30 comment Issue with Spivak's Solution @Amad27: No, it depends on the condition. Consider the function $f(x)=x$, and let $A=\{x\in\Bbb R:f(x)<0\}$; then $\sup A=0$, but $0\notin A$, because $f(0)=0\not<0$.