All Questions

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Set theory with multiple countable infinities

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
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Notation used in this Ring theorem

Lemma: F is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ means? Thanks in ...
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Showing that $\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$

Showing that (1) $$\sum_{n=0}^{\infty}\frac{2^{n+2}}{{2n\choose n}}\cdot\frac{n-1}{n+1}=(\pi-2)(\pi-4)$$ see here (2) ...
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Where is $x^x$ continuous?

The idea of continuity of a function is something I come across quite regularly, but I've never really understood it well. I'm trying to fix that by looking at some interesting functions. What ...
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Simple but hard 2 by 2 system in $x$ and $y$

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases}$$ I mean, other than brute-force.
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Does there exists an additive group homomorphism between two $K$-vector space that is not $K$-linear

My question is: Give me a field $K$. Can we always find two $K$-vector space $V_{1}$, $V_{2}$ and a map $f:V_{1}\rightarrow V_{2}$ such that: (1) If we view $V_{1}$, $V_{2}$ as additive group, then ...
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How to prove that linear transformations transforms lines into lines

I want to answer this: Given $T(x,y)=(ax+by,cx+dy)$ prove that: T transforms straight lines into straight lines T transforms parallel lines into parallel lines I have no clue in ...
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What is a transitive relation on set S

MY answer: Given r,s,t$\in S$ a transitive relation on the set $S$ is when the elements $rRs$ and $sRt$ then $rRt$ i.e., $rRs\land sRt\rightarrow rRt$ Does my definition words correct?
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Calculate $\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$

I'm trying to find this limit $$\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$$ From the graph I can see that it equals $1/2.$ I've looked into making substitution in order to modify the ...
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How to find $\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$?

If n is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
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Finding a bijection and using the Schröder-Bernstein to prove same cardinality

I've been asked to prove that $\mathbb{R}$ and the interval $(-\infty,0)$ have the same cardinality using two methods, one being find a bijection and the other to use the Schröder-Bernstein theorem. ...
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Logic of Set Theory & Partially Order (Informative Discussion)

My final exam passed but, honestly I want to understand what this (Question 4) problem means because I don't know what it is asking for. I am a undergraduate, so it would be most helpful if the ...
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Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
I'm trying to resolve $$f(x,y)=\int_{0}^{1}\int_{0}^{2}e^{-xy}dxdy$$ with Gauss- Legendre quadrature and Simpson's rule, but being honest I have no idea how to write it on MATLAB. So, that's my ...