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1m
comment Is the extended real line a metric space?
Ah yes... that's it!
1m
comment Four definitions for Borel algebra in $\mathbb{R}$?
Can you please make your question more clear. It seems that you are having trouble because you have seen four different definitions of a Borel algebra in $\mathbb{R}$? The appropriate way to ask a question then would be, "How are the following four definitions equivalent: [then list them]".
17m
comment Is the extended real line a metric space?
Also, it is possible to make the extended reals a metric space. You have to define a metric using an integral and the log function... but I can't remember the details off of the top of my head. In particular, the distance between $-\infty$ and $\infty$ is actually finite, so it defines a proper metric.
26m
reviewed Approve Likelihood at least 2 out of $n$ numbers are visible to each other in $\mathbb{Z}^n$
26m
reviewed Approve Rolle's theorem question
35m
comment Is the extended real line a metric space?
Extended real numbers are weird.... Because we allow $\infty$ to be in this number system it doesn't seem too strange that every sequence is bounded since after all $-\infty \le r \le \infty$ for every $r\in\overline{\mathbb{R}}$. So I would say that this sequence is bounded in this regard.... but ask your professor since this is kind of a technical question.
11h
comment Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
No. You are looking for $b$, correct? Well I gave it to you....Explicitly $\lambda = \frac{5}{|a|}$ and I said that $b=\lambda a$, therefore $b=\frac{5}{|a|} a = \frac{5}{|a|} (3,2,-1) = (\frac{5}{|a|}\cdot 3,\frac{5}{|a|}\cdot 2, \frac{5}{|a|} \cdot (-1)) = (\frac{15}{\sqrt{14}} , \frac{10}{\sqrt{14}}, \frac{-5}{\sqrt{14}})$.
11h
comment Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
If a vector $x$ is in $\mathbb{R}^3$ then we can label it with components $x=(x_1,x_2,x_3)$. The rule for scalar multiplication by a number $\lambda$ is, $\lambda x = \lambda(x_1,x_2,x_3) = (\lambda x_1,\lambda x_2,\lambda x_3)$. So think about how you can apply this to my solution... what's your $\lambda$ in this case?
12h
comment Pre-image of $f(x,y) = xy$
The possibilities for $1$ or $-1$ are pretty easy, right? When $y=\frac{1}{x}$ or $y=-\frac{1}{x}$.... so then after a moment's thought you will realize that the points that output values between $-1$ and $1$ will be forced to lie between the curves $y=\frac{1}{x}$ and $y=-\frac{1}{x}$ since these are bounds on the values.
12h
comment Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
Do you know how scalar multiplication works?
12h
comment Sequence of equations
I put a hat on it with a zero... hopefully you like it more now
12h
revised Sequence of equations
added 7 characters in body
12h
comment Pre-image of $f(x,y) = xy$
Do you want a demonstration of why it continuous? Then Euler88's comment about nets (or sequences) shows it's continuous... therefore his "answer" shows that $f^{-1}(U)$ is open. Otherwise, consider my answer, and ask yourself why that set is open (this isn't difficult either)... just pick a point in the region I specified and clearly you can find an open ball around it that is contained in that region
12h
answered Pre-image of $f(x,y) = xy$
12h
comment Pre-image of $f(x,y) = xy$
If that's "exactly" what the question is asking.... ask it in your question above. It's becoming unclear what your actual question is
12h
comment Pre-image of $f(x,y) = xy$
They may not be allowed to use general results / definitions from topology....... Oh snap... it's tagged under topology
12h
comment Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
Even according to the question / solution you posted you see that there are many choices for your solution.
12h
answered Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
12h
comment Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$
You need to read more carefully what the definition of $\text{com}_a(b)$ is.
1d
comment How many ways are there to solve a Rubiks cube?
Definitely smaller than this. Consider a layout of streets where you begin at one point and need get to a destination. If we restrict ourselves to a certain number of streets (say only downtown) then we know there are optimal solutions... and really bad solutions, but any solution that involves returning to the same intersection of streets is unacceptable because this is not "non-repeating", so this automatically forces us to exclude paths that will force us back to a previously visited position. Not sure if this is what the OP is asking or if this analogy holds for the R. Cube though...