2,482 reputation
314
bio website
location
age
visits member for 2 years, 3 months
seen Dec 10 at 18:04

Jul
19
comment What is basis of $\mathbb{R}$
A single open interval won't be enough, no. (If you take the open interval (0,1) as a base for your topology, (2,5) isn't open, but it should be.) You can take the set of all open intervals, though. Or you could take the set of all unions of two open intervals. Or you could take all open sets. There are many bases.
Jul
18
comment Extensions of number fields
Finite integral domains are fields.
Jul
18
comment How to describe the family $\tau$ of all open sets of $(\mathbb R^2,\delta)$
$B(p, \varepsilon)$ should only really be defined for $\varepsilon > 0$, so you can throw that problem away immediately. Otherwise, yes, you're right. All $\delta$-open balls are a singleton set (but not $\{0\}$), or a Euclidean open ball around zero, or a union of the two. So any set that doesn't contain zero is open; and any set that contains zero and a small Euclidean open ball around it is open. A non-open set would be, for example, $\{(x,y) : y \leq x^2\}$.
Jul
18
comment Is a graph with only one node a connected graph?
Yes. (It's either connected or disconnected...)
Jul
17
comment Prove that at a party with at least two people, there are two people who know the same number of people…
@CptSupermrkt It looks like a proof to me. Proofs don't need to be formal, they simply need to be watertight. ;)
Jul
17
comment Help in a proof in Hungerford's book
Suppose u is a unit. Then there is some v such that uv = 1. Apply deg to both sides of that equality, and work out deg u.
Jul
16
comment Proof that for any interval (a,b) with a<b in the real numbers contains both rational and irrational numbers?
Okay, will do...
Jul
16
comment Can we conclude any information about the isomorphism classes of groups of order $n!$?
I mean I computed the first few, inputted it into the OEIS, and it told me that the 7th term "seems to be unknown". How out-of-date that is I don't know, and it doesn't provide its sources, so nor do I. :) oeis.org/A133777
Jul
16
comment Impossible identity? $ \tan{\frac{x}{2}}$
Paul: would you have written $\frac{x}{2} = \tan^{-1}(t)$? If so, you're almost there. Look at the graph of $\tan$, and notice that adding $\pi$ doesn't change anything (i.e. $\tan(x/2) = \tan(x/2 + \pi)$ - so we can't quite work out which $x/2$ this $t$ came from, but we can work it out up to adding or subtracting a few $\pi$s.
Jul
16
comment Impossible identity? $ \tan{\frac{x}{2}}$
(@PaulthePirate: you might find it helpful to latch onto a particular example. One I found helpful: suppose you're driving in your car, and you are at position $x$ at time $t$, then your velocity is $dx/dt$ ("the change in $x$ with respect to $t$"). Of course, $x$ varies with $t$, not vice-versa (you may revisit the same position at many different times, but not vice-versa). (If you drew a graph, you'd probably draw $t$ on the horizontal axis and $x$ on the vertical axis, showing that $t$ is somehow varying of its own accord and $x$ depends on $t$, right?))
Jul
16
comment Can we conclude any information about the isomorphism classes of groups of order $n!$?
For n = 1, 2, ..., 6, the number of groups of order n! is 1, 1, 2, 15, 47, 840. The 7th term seems to be unknown. You may also like this: icm.tu-bs.de/ag_algebra/software/small/number.html . I couldn't find anything more general.
Jul
16
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@fhyve Right - I guess I was just trying to hammer home the point that any structure that can be imposed on one can be imposed on the other (for obvious but somehow stupid reasons), but that that new structure isn't necessarily interesting. I can give [0,1] a few topologies and algebraic operations at random, but (it seems to me that) this isn't interesting because the structures don't interact. Imposing the structure of $\mathbb{C}$ on $\mathbb{R}^2$ somehow isn't interesting in the same sort of way. You get two multiplications that never talk to each other, for example.
Jul
16
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@Cancan No problem! Sorry if this was too theoretical - I got a little carried away. ;)
Jul
15
comment Determine whether the function is a linear transformation:
Read "T(u+v) must equal T(u) + T(v) for all u and v in your space", etc.
Jul
15
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
I agree with Arkamis. The limit definition may or may not be confusing, but the point is that, by applying the 'same' definition to $\mathbb{R}^2$ and $\mathbb{C}$, you get different things.
Jul
15
comment What are the left and right zeroes of the binary union operation?
What have you tried? (If you're not sure what to do, try playing with U = {a, b, c}, for example.)
Jul
15
comment Show that every finite group of order $n$ is isomorphic to a group consisting of $n\times n$ permutation matrices.
Cayley's theorem doesn't say that $P_n$ is isomorphic to $S_G$. ($G$ is your group of size $n$. $P_n$ has size far bigger than $n$, though. Cayley's theorem says it's isomorphic to some subgroup of $S_{P_n}$, which isn't what you wanted to show.) You'll need to show that directly (which you promised to do in the attached image, but didn't seem to do - you gave an example of it when n = 4).
Jul
15
comment Which Lie group / algebra is generated by these three matrices?
They do not form a Lie algebra on their own. (They don't even form a vector space.) But this isn't what the word "generate" means to my mind. "What Lie algebra do A, B and C generate?" means "what is the smallest Lie algebra containing A, B and C?". There is definitely a Lie algebra containing A, B and C (e.g. $\mathfrak{gl}_n$), and so there is definitely a smallest Lie algebra (take all the Lie algebras containing A, B and C, and intersect them). Consider an analogy: three vectors don't necessarily form a vector space, but they can generate (span) one. Likewise three generators of a group.
Jul
15
comment How many positive integers $n$ satisfy $n = P(n) + S(n)$
"If there cannot be three digit number, there cannot be higher cases." Why? (I'm not saying you're right or you're wrong, but this isn't an argument in itself.)
Jul
15
comment Which mathematical tool or method should I use to compare two matrices most efficiently?
It's not really clear what you mean. Can you give an example?