# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### Linear system of equations over $\mathbb{Z}_7$

I had the following set of simultaneous equations in $\mathbb{Z}_7$. $$3x+5y=1$$ $$4x-5y=5$$ Now adding them we get $$7x=6$$ And this has no integer solution in $x$ since $7$ and $6$ are ...
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### How to find the variance of a normal distribution?

X has normal distribution with the expected value of 70 and variance of σ. It is known that $P(67.36\le X \le 72.64) = 0.34$ find σ So if I understand this right we know that ...
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### Prove that if v ≥ 11 , then G and G′ cannot both be planar.

Question is here So, for the last part (v), I used the answer from (i), which is e<1/2(v)(v-1) and the condition: e<3v-6. I equated the two equations and say graph G can only be a planar when ...
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### Twin prime conjecture proof error

I am absolutely sure this is wrong but I can't find why. For every integer $n$ there exist a finite number of primes less than $n$. Take the set containing those primes and multiply them together to ...
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### Probability that $A \cup B$ = S and $A \cap B = \phi$

Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing$ is? My attempt Total number of cases= ...
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### Show two tangents to a parabola through a point on directrix are orthogonal

Given is: ..a point P on the directrix of a parabola with foci $F$. What I want to show is: the tangents of a parabola given by $y=kx^2$ through $P = (x_0, y_0)$ are orthogonal and the line between ...
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### Show that $\phi(N) \leq H$

Let $\phi: G \to H$ be a group homomorphism and $N \leq G$ (with $G$, $N$ and $H$ groups). Show that $\phi(N) \leq H$ So this is what I did: Obviously $\phi(N)$ is a subset of H because $N$ is a ...
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### Is it weird to say $A \in B \in C$?

I've just noticed that I've never seen any text say $A \in B \in C$, which is why when writing it myself it immediately looked weird. For context, I was proving a result about the topology ...
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### Calculating the probability using Poisson Distribution

During working hours, an office switchboard receives telephone calls at random and at an average of 3 calls per minute. a) What is the expected number of calls received during a five minute ...
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### The image of an injective function whose domain is a topological space also a topology

Let $(X, T )$ be a topological space, and let $f : X → Y$ be an injective (but not necessarily surjective) function. QUESTIONS. (1) Is $T_f := \{ f(U) : U ∈ T \}$ necessarily a topology on $Y$ ? ...
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### If $m$ is a positive integer, show that $(ma, mb) = m(a, b)$ .

What I did was let $(a,b)=d$. Then writing the linear combination, $max+mby=md$. Then, to prove that any common divisor of $ma$ and $mb$ can divide $md$. I let $ma={ma_1}{c}$ and $mb={mb_2}{c}$. Then, ...
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### Suppose $x_n$ is a sequence of positive monotonically increasing random variables converging to $X$. Show $\lim_{n \rightarrow\infty}E(x_n)=E(X)$

I am hoping to get some verification of the below proof. I am worried that I am missing something conceptually. That $\lim_{n\rightarrow \infty}E(x_n)\leq E(X)$ is clear since $E(x_n)\leq x$ for any ...
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### Finding the radius of a sphere inscribed in a right prism

We have right prism $ABCA_{1}B_{1}C_{1}$ and points $E$, $D$ such that: $A_{1}E:EB_{1}=B_{1}D:DC_{1}=1:2$ The distance between lines $AE$ and $BD$ is $\sqrt{13}$. Find the ...
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### Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$.

Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$. So my textbook divides the problem into first case $p=1$ and integrates and cases $p<1$ in which it uses integration by ...
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### Proof of the negation of an existential Quantifier

This was an exercise in my book and I was wondering if I got it correct. ...
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### Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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### Least upper bound property of decimal representation of reals

This is my attempt at a proof that real numbers represented by infinite decimals satisfy the least upper bound property, i.e. every upper bounded set has a least upper bound. I am not sure it is ...
I'm given that $\varphi = \arctan\left(\frac{y}{x}\right)$ and I'm asked to show that $$\frac{\partial x}{\partial \varphi}=-r\sin\varphi$$ I've tried to do this and I'm pretty sure this isn't true. ...