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seen Apr 13 at 16:35

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
awarded  Mortarboard
Jul
16
awarded  Good Answer
Jul
16
answered Proof that for any interval (a,b) with a<b in the real numbers contains both rational and irrational numbers?
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 How out-of-date that is I don't know, and it doesn't provide its sources, so nor do I. :) oeis.org/A133777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
awarded  Nice Answer
Jul
15
answered Determine whether the function is a linear transformation:
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
answered Show that every finite group of order $n$ is isomorphic to a group consisting of $n\times n$ permutation matrices.
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
answered What's the difference between $\mathbb{R}^2$ and the complex plane?