# Tagged Questions

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### Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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### Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
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### The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
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### Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
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### elegant proof of Radon–Nikodym theorem

Do you know about an elegant proof of Radon–Nikodym theorem, which is not as cumbersome as the usual ones?
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### How we got $z\cdot(x+y)=x\cdot y$

This is from "Test of math at 10+2 level": A vessel contains $x$ gallons of wine and another contains $y$ gallons of water. From each vessel $z$ gallons are taken out and transferred to the other. ...
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### A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
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### If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
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### If $G$ is disconnected and the vertices $x,y$ are adjacent in $G$, then there is a vertex that isn't adjacent to $x$ and isn't adjacent to $y$.

I'm just starting graph theory and I'm trying to prove the following: Let $G$ be a simple disconnected graph with vertex set $V(G)$ and edge set $E(G)$. If $x,y\in V(G)$ and $xy \in E(G)$, then ...
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How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $>2$ This is a problem I made ...
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### Proving or Disproving the Sum in a Set

I am doing review questions for an exam and I am completely stumped on this particular question: Let A = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}. Prove or disprove that if I select 10 distinct ...
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### $x^2\mid27 \implies x\mid9$ : Prove

$x$ is given as a natural number. I was trying this by direct proof: assume $27\mid x^2$, then $x^2 = 27m \Longrightarrow x=3\cdot \sqrt{3m} \Longrightarrow \sqrt{3m}$ must be integer ...
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### Verification of a proof involving Hausdorff max. principle and collections

I am given this proposition to prove which is a corollary to HMP(Hausdorff maximality principle). My two concerns are: 1. is my attempted proof correct? and 2. Is HMP applicable even when we're ...
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### If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$.

In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that: If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$? I'm not familiar with these kind of calculations, so I'd like ...
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### Justifying exchange of limits in a double sum - a dubious proof

It is a standard theorem (given in Rudin's Principles of Mathematical Analysis, page 175, and many other places), that if $\{a_{ij}\}$ is a doubly indexed sequence, and \sum_{j=1}^\infty |a_{ij}| = ...
### Is this a good proof of Wilson's theorem? — ($(n-1)!+1 \equiv_n 0$ iff n is prime)
Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime. To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p: ...