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Mar
27
comment How to prove that every number, such that its square root is an integer, has an odd number of divisors?
why must an odd product imply an odd number of divisors?
Oct
15
comment Clarification for proof of $\mathbb{Q}$ being dense in $\mathbb{R}$ (Rudin's PMA)
@ThomasAndrews And if it is the $nx$ obtained from $n(y-x)>1$, how does one know to focus on $nx$ and not $ny$?
Oct
14
comment Clarification for proof of $\mathbb{Q}$ being dense in $\mathbb{R}$ (Rudin's PMA)
@ThomasAndrews When you first mention $nx$, are you referring to the $nx$ obtained from $n(y-x)>1$?
Oct
14
comment Baby Rudin Theorem 1.20 (b) Proof
@BrianM.Scott Why doesn't the set begin with "-m_2"?
Jun
26
comment Showing path connected matrices of a group $G$ is a normal subgroup
In the first part, to allow for the inverse of $B$, do I need $\tau^{-1}$ to be continuous too or is it enough to say $\tau$ is continuous? Is this equivalent to saying if $B\in H$, we want to show $B^{-1}$ is in $H$? And you don't need to edit the post just to fix $I$, it's clear now.
Jun
17
comment Open sets in $\Bbb{R}^2$ and $\Bbb{R}^3$
So the unit intveral on the x-axis wouldn't be open in $\Bbb{R}^2$ right?
Jun
17
comment Open sets in $\Bbb{R}^2$ and $\Bbb{R}^3$
@Berci Thanks for that
Jun
17
comment Open sets in $\Bbb{R}^2$ and $\Bbb{R}^3$
Yes, but it is still a union of open balls. So in some sense, are open balls the basis of all open sets in $\Bbb{R}^2$?
Jun
17
comment Topologist's sine curve is connected
Doesn't disconnected mean that there are two disjoint open sets $A$ and $B$ such that the entire set $S$ is equal to $A\cup B$? But when you take your definitions of $A$ and $B$, the union of them is more than what is needed, no?
Jun
9
comment Can a group of order $55$ have exactly $20$ elements of order $11$?
How do you get that there would be 34 elements remaining with order 5?
Jun
9
comment Showing a compact metric space has a countable dense subset
I'm struggling with why $d(x,y)$ has to be less than $1/n$. If the definition of a dense set just requires that another point of $D$ lie in some open neighbourhood of $x \in X$, couldn't it be possible that $y$ is still in $B(x,\epsilon)$ but that the distance between the two points is not less than $1/n$?
Jun
8
comment What can we say about the order of a group given the order of two elements?
To generalize, do I just take all divisors of $n$? If every nontrivial element has the same order, then it must be a $p$-group because the only divisors of $p$ are $1$ and $p$ itself (this means the group $G$ in my problem is certainly not a $p$-group). Is this correct?
Jun
7
comment What can we say about the order of a group given the order of two elements?
could we say it has an element of order 2, 3, and 5? $x^6 = (x^2)^3 = (x^3)^2$, and similarly for $x^{10}$. I know that doesn't say much about $G$, but I'm just trying to figure out what more can be said.
Jun
7
comment What can we say about the order of a group given the order of two elements?
@MartinArgerami Aren't they saying the same thing? Sorry, I was trying to be more formal about it. Thanks for pointing it out
Jun
3
comment $f(x) = \frac {e^{2x-1}} {(1+e^{2x-1})}. $ What is the value of $ f(1/2009) + f(2/2009) + … + f(2008/2009) $?
@CalvinLin If you don't mind me asking, what is the difference between brilliant.org and sites such as art of problem solving?
Jun
3
comment solve system of linear congruences mod 13
Ah, okay. Thanks. Btw, when you multiply $-5$ to equation $I$, did you mean to write $-5$ on the RHS?
Jun
3
comment solve system of linear congruences mod 13
In my textbook, multiplication by a scalar m on $a \equiv b \pmod{c}$ gives $am \equiv bm \pmod{mc}$. Why is it that you don't multiply the modulus number $13 $in the beginning of your solution?
Jun
2
comment Showing existence of an element with order $p$
It's because the order of an element of a group has to divide the order of the group. The only possible divisors of $p^n$ are $1,p^2,\cdots, p^n$. The only element that has order 1 is the identity $e$. So we just pick an element not equal to $e$. Is this correct?
Jun
2
comment Has anyone studied this operator?
@KeyIdeas I had read one of your posts about Gauss's Algorithms I was wondering if it was called something else officially. I'd simply like to read up on it, but I can't find its wikipedia page.
Jun
2
comment Showing existence of an element with order $p$
Sorry, I was referring to how I need to pick that $a$ such that the order is one of $p^1, \dots, p^n$.