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Mar
21
answered Avoiding proof by induction
Mar
20
comment non-commutative infinitesimal extension of $\mathbb R$
@QiaochuYuan, I know, this is just an issue of terminology. I'm just curious about the analogy and whether or not it indicates that $\beta(\mathbb R)$ can be though of as a nonstandrad model of $\mathbb R$, but not in the usual technical sense of nonstarndard (i.e., transfer does not apply).
Mar
20
asked non-commutative infinitesimal extension of $\mathbb R$
Mar
18
comment Does this imply that $F_1$ and $F_2$ are isomorphic as fields?
The uniqueness of finite fields says any two finite fields of the same size are isomorphic, so an isomorphism exists. The proof is classical but it does not proceed by concocting an isomorphism from an additive iso and a multiplicative iso. Is there any reason you don't just study the standard proof of the uniqueness of finite fields?
Mar
18
answered Does this imply that $F_1$ and $F_2$ are isomorphic as fields?
Mar
17
comment connected and path connected
yes, since a path connected space must be connected (through the non-trivial fact that $[0,1]$ is connected).
Mar
16
comment Why does Totally bounded need Complete in order to imply Compact?
Because you can very easily cover $\mathbb Q\cap [0,1]$ by finitely many intervals of any prescribed length $r>0$.
Mar
15
revised Discrete Mathematics: Relations
deleted 2 characters in body
Mar
15
answered Discrete Mathematics: Relations
Mar
15
comment How to prove that homology is a functor?
$Ab$ is the category of abelian groups. Its objects are abelian groups. The homology construction associates a certain quotient group with every object of $cKom$.
Mar
15
answered How to prove that homology is a functor?
Mar
15
comment Why is $D^n/\sim$ homeomorphic to $\mathbb{RP}^n$?
it tells you that you should review what quotient spaces are.
Mar
15
comment Why is $D^n/\sim$ homeomorphic to $\mathbb{RP}^n$?
can you think of a function from the set of lines you describe to the set of pairs of points in $D^n$? (hint: intersection). Are these points always antipodal (hint: yes). What does that tell you?
Mar
15
comment Why is $D^n/\sim$ homeomorphic to $\mathbb{RP}^n$?
but you didn't specify the topology.
Mar
13
answered For sets $A$, $B$, and $C$, why is $A\times B\times C$ is not the same as $(A\times B)\times C$.
Mar
12
comment Confusion about notation for a metric space.
yes, it's given by the absolute value function $d(x,y)=|x-y|$. Though what your professor intends exactly is unclear. Did you see a construction of the reals?
Mar
11
revised If $X^TX=0$ show that $X=0$
edited body
Mar
11
comment If $X^TX=0$ show that $X=0$
What did you try? Try it first for a 1x1 matrix. Then 2x2. See any pattern?
Mar
11
comment Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S
Lagrange's theorem is so elementary and so easy to prove that one should ask why would you try to find any proof that does not use it, if you have a proof that does.
Mar
8
comment How do I prove that a boolean function constructed using $\vee$ and $\wedge$ (without using $\thicksim$ ) must attain the value 1 at least once?
you need to prove that given any boolean function of that form, it must attain the value 1. So, firstly, you can't choose the boolean function. Secondly, you need to review your understanding of the material, since when you plug in $0$'s you certainly don't get $1$. Enjoy working on your homework!