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comment When can I divide both sides of an equation if one side is zero
For equality, you can multiply both sides by the same number (and therefore also divide by any non-zero number since that's just multiplication by the inverse of the number). Whether the number is positive or negative doesn't matter. It does however when you have an inequality because if you multiply by a negative number, you have the inverse the inequality.
Apr
17
comment Power set equinumerosity. Is this proof correct?
$H(B)$ isn't defined. $B\in \mathcal P(B)$ which is the codomain of $H$. And to prove that a function is the inverse of another, you just prove that when you compose them (on both sides), you get the identity.
Apr
17
comment Power set equinumerosity. Is this proof correct?
You didn't define $H$, you only gave its domain and codomain. $H:X\mapsto \{f(a) / a \in X\}$. And I think it'd be easier to build its inverse and prove that it is indeed its inverse instead of proving that it's bijective.
Jan
23
comment $x=yx$. Can this statement be true when we don't know that $y=1$?
$yx=x \implies yx-x = 0 \implies x(y-1)=0 \implies x= 0 \text{ or } y = 1$
Jan
21
comment Finding $|a|$, a complex number, given a system of equations
Have you tried using $z\overline{z}=|z|^2$?
Jan
18
comment [Verification]$G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
a^n or a^{stuff}${}{}{}{}{}{}$
Dec
27
comment Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail?
The idea is that the roots are the eigenvalues of the companion matrix, and that we know how to computer the eigenvalues of a matrix numerically.
Dec
27
comment Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail?
See mathworld.wolfram.com/PolynomialRoots.html
Dec
20
comment Derivative of a quadratic form
Have you tried looking at an example? Like $X=\begin{pmatrix}0&i\\-i&0\end{pmatrix}?$
Dec
14
comment Basic Axiomatic Definitions in Categories and Allegories (Freyd and Scedrov)
For example $x\mapsto \cfrac{1}{x}$ isn't defined at $0$ so it's not total (as a function of $\mathbb R \to \mathbb R$). And a binary partial operation is to a binary operation what a partial function is to a function.
Dec
14
comment Basic Axiomatic Definitions in Categories and Allegories (Freyd and Scedrov)
And a binary operation is a function $f:A\times A \to A$. You can also represent a function $f:X\to Y$ by a subset $F$ of $X \times Y$ so that $\forall x\in X,\forall y_1,y_2 \in Y, (x,y_1)\in F \land (x,y_2)\in F \implies y_1 = y_2$ ([functional] given an input, you have at most one output) and $\forall x\in X, \exists y \in Y, (x,y)\in F$ ([total] given an input, you have at least one output). A partial function is like a function in the sense that it is functional but it doesn't have to be total. In other words, a partial function is a function that can't give an output on some inputs.
Dec
14
comment Basic Axiomatic Definitions in Categories and Allegories (Freyd and Scedrov)
He might represents an object $A$ by $id_A$, then it all kind of makes sense. But then, it's weird that he doesn't simplify the rules by to get, for example, $\square(xy)=\square x$ instead of $\square(xy)=\square( x(\square y))$
Dec
12
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Dec
10
comment How many graphs on the vertex set {1, …, 12} have exactly 15 edges?
The total number of posssible edges of a graph with $n$ vertices is $n^2$ if you allow edges from an edge to itself and $n^2 - n = n(n-1)$ otherwise.
Dec
7
comment Prove $\{g(x_n)\}_{n=1}^\infty$ converges
One way to work you way through this kind of problem is to rename everything that's small and positive to $\varepsilon_1$, $\varepsilon_2$, etc. Then, you just have to find out which ones must be equal.
Nov
30
revised A function whose graph has vertical asymptotes at $x=+2$ and $x=-2$, and a horizontal asymptote at $y=0$
added 11 characters in body
Nov
30
comment Can we build infinite products in $k[[X]]$?
What would you want it to be for $P\equiv -1$?
Nov
30
comment Is there any diffeomorphism from A to B that $f(A)=B$?
You probably tried the function $f(x)=e^{i\pi/2}x^3$. And you have the problem that at $f'(0)=0$. I suggest you try $g(x)=f(x-c)+e^{i\pi/2}c^3$ to try to push the problem outside of the domain. The only difference with $f$ is that $g$ turns around $c$ instead of turning around $0$.
Nov
27
comment proof that $a_n$ is a null sequence
That's overkill.
Nov
22
comment When does this complex series converge?
For some $x$ expressed in terms of $z$, you series is $\sum\cfrac{x^n}{n}$ for which you should be able to answer the question.