# All Questions

1,006,278 questions
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### topology and metrizability

Let X be a nonempty set and Y is a proper subset of X. Set τ =2^Y U {X}. a) Show that τ is a topology on X. b) Find all nonempty subsets X such that ...
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### How to solve propositional logic problems by induction

I'm trying to solve a bunch of problems like this one and every time I get stuck. So I don't need an actual solution but to understand how you solve this kind of problems. I know they're usually ...
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### I don’t understand this,can you explain it simply

What is the condition that the cubic y=ax^3+bc^2+XX+d shall have two extremes?
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### Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers. For example: The commutative axiom x.y = y.x The distributive axiom x.z + y.z = (x+y).z Well, the thing is that we are working ...
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### Proper class if and only if there is a bijection with the class of ordinals.

I am looking for a way to formalize the sentence "$A$ is a proper class", and so I was wondering if I could write it like "a proper class is something that is in bijection with the class of all ...
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### Branch Points and Branch Cuts for cube root

Having trouble with finding the branch points/branch cuts of this function: f(z)= $3\sqrt\frac{(z-2)(z+1)}{(z+2)}$ ive tried using the equation $z=re^{i\theta +2\pi n}$ but then I don't really know ...
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### Solving $\cos(t\cos\theta)\cos(t\sin\theta) \sin 2\theta = \sin(t\cos\theta)\sin(t\sin\theta)$

I am trying to find all $\theta\in\mathbb{R}$ such that $$\cos(t\cos\theta)\cos(t\sin\theta) \sin 2\theta = \sin(t\cos\theta)\sin(t\sin\theta)$$ holds for all $t\in\mathbb{R}$. Just by looking at the ...
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### Reduction formula for integral $\int 1/(a^2+x^2)^n dx$

I want to solve the integral $\int 1/(a^2+x^2)^n dx$. I guess this can be solved using the reduction formula as in the case of $\int x^2/(a^2+x^2)^n dx$. However, I am not able to do it.
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### How to show that the following sets are equinumerous

This is my first question on this website, so please do give me feedback where nessecary. I got stuck on a question in Computability and Logic 5th edition by Boole i.a. It's question 2.10 and the ...
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### Limit with 3 variables approaching infinite

I'm having some problems resolving this limit. The fact is that I'm finding little information about limits in three variables online and in my manuals, and even less information about three variables ...
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### Find m(A), where m denotes Lebesgue measure

Let A be the set of numbers in the interval [0, 1] that have the digit “0” in the first, second or third place in their decimal expansion. Find m(A), where m denotes Lebesgue measure. My knowledge:...
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### Hey need help . . . . . . . . . .

can someone help me solve this 2 question 1:Let X and Y be sets. Prove that X × Y = Y × X iff X= ∅ or Y= ∅ or X=Y 2:Let X,Y,Z be sets. Prove or refute: • X ∪ (Y × Z) = (X ∪ Y) × (X ∪ Z) • (X × X)(Y ...
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### Statistic Question about Neurons

Question Hi, I want to solve this question and I know what variance, mean, covarianceö however, I am struggling to understand what question asking for. Can anyone help me to clarify what is expected ...
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### Proving the convergence of $\int e^{x^2/2} dF_{1,2}(x)$ in Cramers proof of the

In Cramer 1936 the proof of the Cramer decomposition theorem contains proving the following integrals are finite $$\int e^{x^2/2} dF_{1}(x), \quad \int e^{x^2/2} dF_{2}(x)$$ to later use in finding a ...
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### $\int_0^1\frac{x^2\ln x}{\sqrt{1-x^2}}dx$

$\int_0^1\frac{x^2\ln x}{\sqrt{1-x^2}}dx$ I tried putting $x=\sin \theta$ and changing the limits from $0$ to $\frac{\pi}{2}$and i got $\int_0^\frac{\pi}{2}\sin^2\theta\ln \sin \theta d\theta$ i ...
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### Let $M$ be a $3\times 3$ matrix satisfying $M^3=0$

Let $M$ be a $3\times 3$ matrix satisfying $M^3=0$. Prove that $$\left|\frac{M^2}{2}\pm M+I\right|\neq 0$$
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### Nonsingular projective curve corresponding to $y^2 = x^4+1$

Consider the affine curve $C_1 = V(y^2 - (x^4+1)) \subset \Bbb A^2_k$. In the answers to this question, they claim that there is a (unique?) nonsingular projective curve $C_2$ corresponding to $C_1$ (...
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### Show that $\alpha f + \beta g$ is measurable when $f,g$ are measurable.

Consider a measure space $(\Omega, \mathcal{F}, \mu)$ and two integrable, measurable functions $$f,g: \Omega \to [-\infty, + \infty]$$ I.e., $$\int fd\mu, \int g d \mu \in \mathbb{R}$$ I proved ...
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### Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
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### If C, D, E, and K is matrices, and CKD=E, is there any other way to determine matrix K beside expanding that equation with matrix multiplication?

If C is 3 x 2 matrix, D is 2 x 3 matrix, and E is 3 x 3 matrix, that K should be 2 x 2 matrix if CKD = E. With specific entries of each matrices, I could get the elements of K with expanding the ...
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### Meaning of and /or in the set theory

Winnie has 165 workers in process X,110 in process Y,97 in process Z.Out of this workers 281 are skilled in the activities of x and/or why ,269 are skilled in the activities of y and/or z ,241 are ...
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### Sum of a multiplicative arithmetic function

Prime factorization of $n$ is $\prod p_i^{e_i}$ Then radical of $n$ is defined as $\text{rad}(n)=\prod p_i$ Let $S(N) = \sum_{n=1}^{N}\text{rad}(n)$ I want to calculate $S(N)$ for very large value ...
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### Show that the sequence $y_n = 1+\frac{1}{2^k}+\frac{1}{3^k}+…+\frac{1}{n^k}$, where $n\in \mathbb{N}$ and $k\geq 2$ is convergent. [duplicate]

Let $y_n = 1+\frac{1}{2^k}+\frac{1}{3^k}+...+\frac{1}{n^k}$, where $n\in \mathbb{N}$ and $k\geq 2$. I need to show that this sequence is convergent. I did $y_{n+1}-y_n$ and I got that it is ...
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### Need some help solving this equation

The equation is $$\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$$ I tried squaring both left side and right side then bringing them to same numerator but got lost from there ... any ...
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### Variance random variables

someone can explain me why the difference of 2 random variables indipendent is the sum of 2 random variables and not the difference? So, why Var(X-Y) = Var(x) + Var(y) ?
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### Problem computing Lie algebra of unitriangular matrices

Let $G$ the Lie group of the upper unitriangular matrices, i.e. \begin{align} G : = \{ A= (a_{ij})_{ij} \, \, |\, a_{ii} = 1 \, \, \, \forall \, i \, \, \text{and} \, \, a_{ij} = 0 \, \, \forall \, i&...
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### Bernoulli's inequality and secuences

How can I use Bernoulli's inequality to prove c^{1/n}->1 for $c>0$ ? I have to use $c^{1/n}=1+Xn$. Thank you!
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### Power set in measure theory

Let X be any set and A = P(X) be the power set of X. Let x1,...,xn be distinct points in X and let α1, . . . , αn be positive real numbers. Show the measure on A. I'm uncertain if μ = α1δx1 + α2δx2 +...
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### how the following ways of selection differs from each other

Suppose I want to select $2$ elements out of $6$ elements. Then I get $6C2 = 15$ combinations. Here $C$ represents the standard formula of ($N C R$). Now my question is what differences arise when ...
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### Is it true that $P(A|B) = P(A|C) \cdot P(C|B)$?
I think that $$P(A|B) = P(A|C) \cdot P(C|B)$$ is True. You are just transforming the information from $B$ through $C$. Is this correct and if it is, what's the name for this property?