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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 $f(0)=0\not<0$.