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

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1answer
29 views

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 ...
1
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1answer
16 views

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 ...
-1
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0answers
24 views

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 ...
3
votes
1answer
35 views

Failure of proof for $\operatorname{null}T_1 \subset \operatorname{null}T_2 \implies \exists S~\text{s.t.}~T_2 = ST_1$

In S.Axler's "Linear Algebra Done Right (3ed) an exercise asks of the following: Suppose $W$ is finite dimensional and $T_1, T_2 \in \mathcal{L}(V,W)$. Prove that $\operatorname{null} T_1 ...
4
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2answers
134 views

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
2
votes
1answer
48 views

Trying to prove Cantor-Bernstein-Schröder following these steps

I know a proof for this theorem is a recurrent issue but I've checked wikipedia's proof and several posts in this forum about it and even if I found some similarities I couldn' solve my problem. Let ...
0
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1answer
28 views

Verify $\frac{\cot x -\tan x}{\cos x + \sin x}=\frac{\cos x - \sin x}{\sin x \cos x}$

Verify $$\frac{\cot x -\tan x}{\cos x + \sin x}=\frac{\cos x - \sin x}{\sin x \cos x}$$ After several tries I cannot find a concrete way of proving/verifying this. Any help/hints?
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0answers
27 views

Prove the relation $R = \{(x,y)\in \mathbb{R} \times \mathbb{R}: \text{ |x|< |y| or x=y} \}$ is antisymmetric.

Prove the relation $R = \{(x,y)\in \mathbb{R} \times \mathbb{R}: \text{ } |x|< |y|\text{ or $x=y$} \}$ is antisymmetric. Proof: Suppose $ x R y$ and $ yRx $. Then $|x|<|y|$ or $x=y$. ...
0
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0answers
24 views

On proving that the surjectivity of a function is implied by the existence of a right inverse.

As I've mentioned in a previous question, the definition of a surjective function has been giving me some trouble however, I think that along with the answers to that question, this should resolve ...
0
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1answer
40 views

How to prove the group of roots of unity in $\mathbb{C}$ is a group

I mostly need help with proving $G$ is closed but a verification of the other parts is appreciated. Let $G = \{z \in \mathbb{C} \mid z^n=1$ for some $n\in \mathbb{Z^+}\}$ I want to start by proving ...
0
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0answers
27 views

Proof tree of $[(\phi\lor \psi)\land (\phi \lor \chi)] \to [\phi \lor (\psi \land \chi)]$

I need to construct a proof tree of: $$[(\phi\lor \psi)\land (\phi \lor \chi)] \to [\phi \lor (\psi \land \chi)]$$ Could someone check the following proof tree? I first proved the following: ...
-2
votes
1answer
23 views

Convergence in distribution of $X_1, X_2,… $ to a constant $c$ implies convergence in probability

this is my proof attempt at convergence in probability. Is it right- also if it is right- have we not shown something stronger given we've got the final bit equals 1 and not just tending to 1 so ...
0
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2answers
64 views

How to prove $f^{-1}(f(X)) = X$

Suppose $X \subseteq A$. Will it always be true that $f^{-1}(f(X)) = X$? I am try to prove this problem with either proofs or counterexamples. I have found a counterexample for $f^{-1}(f(X)) ...
11
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2answers
1k views

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 ...
1
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1answer
40 views

Showing that this topology is the same as the product topology on $X \times Y$

Let $\mathcal{T}$ be the product topology on $X \times Y = \{(x,y) : x \in X, y \in Y\}$ generated by the basis $\widetilde{\mathcal{B}} = \{U \times V : U \in \mathcal{T}_{_X}, V \in ...
4
votes
0answers
82 views

Probability of having at least $j$ collisions when tossing $m$ balls into $n$ bins

Suppose that we throw $m$ balls into $n$ bins uniformly and independantly at random. We consider collisions as distinct unordered pairs, e.g., if 3 balls are tossed in one bin, we count 3 collisions. ...
0
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0answers
35 views

Existence of algebraically closed extensions

Just want to check everything is fine. Basically, the point is to first take some polynomial ring with some huge amount of variables so that each can become a suitable root when we project it in the ...
0
votes
1answer
36 views

Assume that ${ a }_{ 1 }<{ a }_{ 2 }$. Show that if there is no $3$-chain in our sequence, then ${a}_{3}$ must be less than ${a}_{1}$

Define a $3$-chain to be a (not necessarily contiguous) subsequence of three integers, which is either monotonically increasing or monotonically decreasing. We will show here that any sequence of five ...
1
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3answers
31 views

Let k $\in Z$, such that k $\ge$ -1. Then $k^2 + 1$ is not divisible by 3.

I had this on the exam a few months ago and I am doing it again just for review. I want to check if I did it right this time. Any comment would be appreciated! Proposition: Let k $\in Z$, such that ...
2
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0answers
23 views

Showing that an intersection of indexed sets is a subset of every individual indexed set

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$. This seems obvious to me from ...
1
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3answers
29 views

Let $G$ be a graph with $n$ vertices where every vertex has a degree of at least $\frac{n}{2}$. Prove that G is connected.

First question, if the problem uses a fraction such as $\frac{n}{2}$, would we round down in case $n$ is odd? As for the actual problem, I'm trying to do this with induction and contrapositive and ...
0
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1answer
15 views

Under what assumptions on φ is Tco-φ a topology

Fix a set X, and let φ be a property that subsets A of X can have. Define Tco-φ = {U ⊆ X : A = ∅, or X \ U has φ } . Under what assumptions on φ is Tco-φ a topology on X? What I think: 1. X\X has φ ...
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0answers
21 views

Discrete Mathematics: Prove Expression Is Even

I'm brushing up for my Discrete Math final exam, and would like to know if the following proof is valid... Prove by direct proof that $a^2-5a+8$ is even for any integer $a$. Proof: By ...
0
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0answers
16 views

$Df(x_0)$ is onto. Then there is whole neighborhood of $f(x_0)$ lying in the image of $f$.

Problem says: Suppose that $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is of class $\mathcal{C}^{1}$ and $Df(x_{0})$ has rank $m$ . This means that $Df(x_{0})$ as a linear map is ...
0
votes
2answers
22 views

How to prove this equality about functions over indexed and intersecting sets?

Let $f:A\to B$ be a map of sets, and let $\left\{X_{i}\right\}_{i\in I}$ be an indexed collection of subsets of $A$. I need to prove that $f\left(\bigcap_{i\in I} X_{i}\right) \subset \bigcap_{i\in ...
5
votes
6answers
111 views

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= ...
0
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1answer
38 views

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 ...
4
votes
0answers
34 views

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 ...
4
votes
2answers
79 views

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 ...
0
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2answers
32 views

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 ...
1
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1answer
33 views

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 $Y$ ? ...
0
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0answers
39 views

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, ...
1
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0answers
35 views

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 ...
1
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1answer
24 views

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 ...
0
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0answers
49 views

Is $\|\mathbf{u} -\mathbf{v}\|\leq \|\mathbf{u}\| + \|\mathbf{v}\| $?

Here's my working: $\|\mathbf{u} -\mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + (- 2\, \mathbf{u}\,\bullet\mathbf{v})$ Since, by the Cauchy-Schwarz theorem, $|\mathbf{u}\,\bullet\mathbf{v}| ...
1
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1answer
36 views

Is $f(n)=\begin{cases} \frac{n}{2}&\text{if}~n~\text{is even}\\ \frac{-n-1}{2}&\text{if}~n~\text{is odd}\end{cases}$ a bijection?

let f define by : $$f(n)=\begin{cases} \frac{n}{2}&\text{if}~n~\text{is even}\\ \frac{-n-1}{2}&\text{if}~n~\text{is odd}\end{cases}$$ I would like to show that $f$ is a bijection from ...
1
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1answer
49 views

Homology of $Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$

I want to compute the homology of $M=Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$. I think I have the answer, but I'm not sure how to make it precise. My approach is to consider the affine cover ...
1
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2answers
27 views

Let$ (X; T_{\text{cocountable}})$ be an infinite set, show that it is closed under countable intersections.

Also give an example to show that $\mathcal{T}_{\text{cocountable}}$ need not be closed under arbitrary intersections. I was looking for some feedback on my proof: $X\setminus\bigcap_{\alpha \in I} ...
1
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1answer
36 views

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 ...
1
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1answer
23 views

Proof of the negation of an existential Quantifier

This was an exercise in my book and I was wondering if I got it correct. ...
0
votes
1answer
80 views

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 ...
0
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0answers
10 views

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 ...
3
votes
4answers
63 views

Partial derivatives for polar coordiantes

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. ...
1
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1answer
81 views

Show: equality of angles in ellipse between foci and external point

first please take a loot at this: Given is an ellipse with foci $F1, F2$ and an external point $P$. Through P I have constructed two tangents to the ellipse. I need to show that: $\angle F1PB1 = ...
1
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1answer
33 views

Different ways to find limit in infinity

I would like to find a limit of the function $f(x)=\frac{x^3+\sqrt{x^6-1}}{4(x-2)^2}$ as $x\rightarrow \infty$ and as $x\rightarrow -\infty$. In the first case, both numerator and denominator goes ...
1
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3answers
38 views

Use direct proof to prove: If $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then $B = C$

I'm interested in knowing if the method I used is correct. I've been teaching myself proofs lately and I am having difficulties with how to approach a problem so any general tips would be awesome as ...
4
votes
3answers
194 views

Proving $\frac{d}{dx}x^2=2x$ by definition

I did the following proof earlier and just wanted conformation as to whether it works. The question was to show $$\frac{d}{dx}x^2=2x$$ by the difference-quotient definition of a derivative, and then ...
0
votes
2answers
20 views

Is this proof about equicardinality correct and/or rigorous? Can it be helped?

Here's the proof than a Cartesian product of two countable sets is countable(the proof is used, for example, in C.Pugh's "Real Mathematical Analysis" with one exception: they prove equicardinality of ...
0
votes
1answer
23 views

Holomorphic functions bounded by other holomorphic functions

Here is a problem I encountered while studying old comp problems for a final. Suppose that $f(z)$ is an entire holomorphic function satisfying $|f(z)| \leq |\cos(z)|$ for every $z$. Show that $f(z)$ ...
1
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0answers
45 views

The Maximum Modulus Principle Applied to the Proof of Schwarz Lemma

I am using the following statement of the Maximum Modulus Principle: Theorem: Let $G$ be a region and let $f$ be holomorphic on $G$. Suppose $\exists~ a \in G$ such that $|f(z)| \leq |f(a)| ~ ...