agent154
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 Sep 29 comment Cubic polynomial and its discriminant So, I suppose the graph should be read as "has complex roots" vs "has solutions". The shaded area is where there are no complex roots, but the white area does have some? Sep 24 comment Prove that $2^n>n^4$ for all $n\geq 17$ It genuinely seems like these types of problems come easy with an inordinate amount of practice, and having knowledge of general techniques. I would have never considered many of those inequalities. Sep 22 comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective I'm afraid that goes way beyond my understanding at this level... I was warned by my prof that trying to find an inverse may end up being too messy, so you've just proven that for me... Sep 22 comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective I don't follow how you converted $x^2+xy+y^2$ as $\frac{3}{4}(x+y)^2+\frac{1}{4}(x-y)^2$ Sep 22 comment Proving that $\alpha:\mathbb{R}\to\mathbb{R}$ where $\alpha(x)=\frac{x^{3}}{x^{2}+1}$ is bijective This seems very strange to me, claiming that it is injective under the constraint that $x=y=0$. Shouldn't $x=y$ no matter what? Sep 21 comment Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$. I was fixated on finite sets.. this is why I was confused. Sep 20 comment Determine whether the function $\alpha:A\times B\rightarrow B\times A$ where $\alpha((a,b))=(b,a)$ is injective and/or surjective @SpamIAm I suppose I missed a step. I intended to say $(b,a)=(d,c)$ as these are the images of $(a,b)$ and $(c,d)$. Sep 20 comment Prove that $(\mathbb{R}\setminus \{0\},\sim):= ab>0$ is transitive. Ah. For whatever reason I had not considered multiplying them together. I had thought of dividing them out, but was getting nowhere that way. Sep 19 comment Show that $A\setminus (B\setminus C)=(A\setminus B)\cup(A\cap B\cap C)$ I vaguely recall this from some years ago -- what is this property/equality/law called? I'm referring to the addition of $B$ and $\overline{B}$ to both sides of a union Mar 3 comment How do I design a generating function to count subsets of distinct objects? $(1+x)^{250}$ looks like it does not take into account the limitations of at most 3 per state. Given your other advice, however, I came up with $g(x)=(1+5x+10x^{2}+10x^{3})^{50}$ Dec 18 comment How does the triangle inequality work for $|x-y|$? Yes, that's what I ended up finding. Never occured to me Dec 17 comment How does the triangle inequality work for $|x-y|$? Nevermind - I found a proof that shows the inequality using $|x+y|$ Dec 17 comment How does the triangle inequality work for $|x-y|$? But my question, though, is how $||x|-|y||\leq|x+y|$? My textbook says (without proving) that $||x|-|y||\leq|x+y|\leq |x|+|y|$. Dec 17 comment How does the triangle inequality work for $|x-y|$? I have here that $||x|-|y||\leq|x+y|\leq |x|+|y|$... I was able to find several proofs of the "reverse triangle inequality", but they all start off with $|x-y|$ instead of $|x+y|$ like in the upper and lower bounds given. How does this proof translate from $|x|-|y|\leq |x-y|$ to $||x|-|y||\leq |x+y|$? Nov 3 comment Finding the matrix of a linear transformation relative to two non-standard bases @Omnomnomnom Unfortunately, all it says is what $T$ is for a given $\langle x,y,z\rangle$. Oct 28 comment If $\mathbb{Z}_m^*$ is cyclic, and $\mathbb{Z}_m^*=\langle\overline{g}\rangle$, is $\overline{g}$ a primitive root? @ThomasAndrews I meant to imply when it is cyclic; as in, when $m=1,2,4$, or is of the form $p^{\alpha}$ or $2p^{\alpha}$ where $p$ is an odd prime. Oct 28 comment What are the generators of $\mathbb{Z}_9^*$? @JyrkiLahtonen OK, then from the table alone, assuming I correct it to use multiplication instead of powers, can I find a generator from it? Or do I have to test each one individually with the method above? Oct 28 comment What are the generators of $\mathbb{Z}_9^*$? @JyrkiLahtonen Wouldn't that mean that every element in $\mathbb{Z}_9^*$ is a generator, save for $\overline{1}$? Oct 28 comment What are the generators of $\mathbb{Z}_9^*$? @JyrkiLahtonen But that seems contrary to what I was taught... but again I may have misunderstood Oct 28 comment What are the generators of $\mathbb{Z}_9^*$? @coffeemath notice the headers in my table. Are the headers wrong? I listed all the elements $n$ such that $(n,9)=1$