 # Pre algebra questions and answers

Recent questions in Pre-Algebra Bailee Landry 2022-05-23 Answered

### I was following along a Y-T video proving that there is no useful measure on the power set of $\mathbb{R}$. The proof is too long to lay out here but I'm certain that most respondents here will recognize it if I just describe it.The proof starts by pulling all the rational numbers from (0,1], creating little `boxes' along the interval. Next construct a set A that contains one and only one point from each box. Then create a sequence, ${A}_{n}$ by adding a rational r from (-1,1), jogging A left and right in (-1,2]. Then we take the countable union of the ${A}_{n}$, and here's where I lose the plot. The claim, setting up the final blow in the proof, is that(0,1] $\subseteq \left\{\cup {A}_{n},n\in \mathbb{N}\right\}\subseteq \left(-1,2\right]$I get the RHS but the LHS looks exactly wrong to me. (0,1] contains all the reals and by construction the term in the middle is devoid of the rational numbers. How can the term in the middle contain (0,1]? istremage8o 2022-05-23 Answered

### How do you solve $\frac{x}{3}<5$ ? Brennen Fisher 2022-05-23 Answered

### Let $\left(\mathrm{\Omega },\mathcal{A},\mathrm{P}\right)$ be a probability space and ${\mathcal{F}}_{i}\subseteq \mathcal{A}$ be a $\sigma$-algebra on $\mathrm{\Omega }$.If $\left({\mathcal{F}}_{1},{\mathcal{F}}_{2},{\mathcal{F}}_{3}\right)$ is independent, does it follow that ${\mathcal{F}}_{1}\vee {\mathcal{F}}_{2}:=\sigma \left({\mathcal{F}}_{1}\cup {\mathcal{F}}_{2}\right)$ is independent of ${\mathcal{F}}_{3}$?In the special case, where ${\mathcal{F}}_{i}=\sigma \left({X}_{i}\right)$ for some random variable ${X}_{i}$ taking values in a measurable space $\left({E}_{i},{\mathcal{E}}_{i}\right)$, we've got$\begin{array}{}\text{(1)}& {\mathcal{F}}_{1}\vee {\mathcal{F}}_{2}=\left({X}_{1},{X}_{2}{\right)}^{-1}\left({\mathcal{E}}_{1}\otimes {\mathcal{E}}_{2}\right)=\left({X}_{1},{X}_{2}{\right)}^{-1}\left(\sigma \left({\mathcal{G}}_{1}×{\mathcal{G}}_{2}\right),\end{array}$where ${\mathcal{G}}_{i}\subseteq {\mathcal{E}}_{i}$ is arbitrary with $\sigma \left({\mathcal{G}}_{i}\right)={\mathcal{E}}_{i}\right)$. From (1) the desired claim immediately follows.So, I wondered whether the same holds in general or why it breaks down. Jerry Villegas 2022-05-23 Answered

### How do you write an inequality and solve given "a number decreased by 8 is less than 21"? Scolfaro2y 2022-05-23 Answered

### Finding out how out how much is 100%I know 14% is 41. So how much is 100%? I know it is simple math, but... you know how it is being out of school for a few years. If at all possible, I'd like to see the equation too. Charity Daniels 2022-05-23 Answered

### Why do I get $0.098765432098765432...$ when I divide 8 by 81?I got this remarkable thing when I divided 16 by 162, or, in a simplified version, 8 by 81. It's $0.098765432098765432\cdots$ , or more commonly known as $0.\overline{098765432}$ with all the one-digit numbers going backwards...except for 1. Yeah, it's missing the 1. One, how do I get this remarkable outcome and two, why is it missing the 1? Jazmine Bruce 2022-05-23 Answered

### How do you solve $\frac{1}{3}x>6$ ? dokezwa17 2022-05-23 Answered

### I thought if the lower and upper bound of a summation are equal, that from that would follow that the sum is always equl to 0. But when trying this in Wolfram Alpha the result is just one iteration of the summation term. Can someone please explain if my hypothesis therefore is wrong?<math xmlns="http://www.w3.org/1998/Math/MathML" "> ∑ j = 0 0 1 = 1 Emely Baldwin 2022-05-23 Answered

### How come ${\left(\frac{n+1}{n-1}\right)}^{n}={\left(1+\frac{2}{n-1}\right)}^{n}$?I'm looking at one of my professor's calculus slides and in one of his proofs he uses the identity:${\left(\frac{n+1}{n-1}\right)}^{n}={\left(1+\frac{2}{n-1}\right)}^{n}$Except I don't see why that's the case. I tried different algebraic tricks and couldn't get it to that form.What am I missing?Thanks.Edit: Thanks to everyone who answered. Is there an "I feel stupid" badge? I really should have seen this a mile a way. istremage8o 2022-05-23 Answered

### A really strange question here in my opinion:Let $A,B$ be two measurable subsets of ${\mathbb{R}}^{1}$. Define $f\left(x\right)=|\left(A-x\right)\cap B|$. Evaluate ${\int }_{{\mathbb{R}}^{1}}fdx$. Here $|\cdot |$ refers to the measure.Here $f$ is clearly non-negative, so I tried using the definition of Lebesgue integral:${\int }_{{\mathbb{R}}^{1}}fdx=sup\left\{{\int }_{{\mathbb{R}}^{1}}sdx:0\le s\le f\right\},$where s are simple functions. But I realised that I couldn't come up with any simple functions.I tried breaking up ${\mathbb{R}}^{1}$ into two parts:${E}_{1}=\left\{y\in {\mathbb{R}}^{1}:y\in A\cap B\right\},{E}_{2}=\left\{y\in {\mathbb{R}}^{1}:y\notin A\cap B\right\}$and then considering the sum${\int }_{{E}_{1}}fdx+{\int }_{{E}_{2}}fdx$and then I'm clueless as to how to proceed. I'm guessing the answer should be something intuitive like $|A\cap B|$. Any help would be appreciated. meindwrhc 2022-05-23 Answered

### Let $\left(\mathrm{\Omega },\mu \right)$ be a $\sigma$-finite measure space. Suppose $1\le p<\mathrm{\infty }$. Consider the cone ${L}^{p}\left(\mathrm{\Omega }{\right)}_{+}$ of positive functions of ${L}^{p}\left(\mathrm{\Omega }\right)$.Is ${L}^{p}\left(\mathrm{\Omega }{\right)}_{+}$ weak-closed in ${L}^{p}\left(\mathrm{\Omega }\right)$ ? Jerry Villegas 2022-05-23 Answered

### How do you solve $2v+1>7$ ? Davian Maynard 2022-05-23 Answered

### How do you solve and graph $20b\ge -120$ ? Jaidyn Bush 2022-05-22 Answered

### Considering $X=\left\{0,1{\right\}}^{\mathbb{N}}$, the space of infinite sequences of 0's and 1's, and the $\sigma$-algebra generated by the cylinders sets $\left[{x}_{1},{x}_{2},...,{x}_{n}\right]$, where ${x}_{i}\in \left\{0,1\right\}$.I want to show that the set of all $x\in X$ such that $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{i=1}^{n}{x}_{i}=\frac{1}{2}$ is in this $\sigma$-algebra.I'm a little bit lost on this one, I'm not sure where to start.I know the definition of a convergent sequence, and I know that I should be able to construct the set of such sequences from unions, intersections and complements of the cylinders.Any idea would be of great help! tinydancer27br 2022-05-22 Answered

### Boy Born on a Tuesday - is it just a language trick?The following probability question appeared in an earlier thread:I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?The claim was that it is not actually a mathematical problem and it is only a language problem.If one wanted to restate this problem formally the obvious way would be like so:Definition: Sex is defined as an element of the set boy,girl.Definition: Birthday is defined as an element of the set Monday,Tuesday,Wednesday,Thursday,Friday,Saturday,SundayDefinition: A Child is defined to be an ordered pair: (sex $×$birthday).Let (x,y) be a pair of children,Define an auxiliary predicate .Calculate I don't see any other sensible way to formalize this question.To actually solve this problem now requires no thought (infact it is thinking which leads us to guess incorrect answers), we just computeNow what I am wondering is, does this refute the claim that this puzzle is just a language problem or add to it? Was there a lot of room for misinterpreting the questions which I just missed? Trevor Wood 2022-05-22 Answered

### How do you solve $3m<4$ ? il2k3s2u7 2022-05-22 Answered

### Writing a fraction as ${x}^{n}$I came across this fraction after practising with bunch for a while. How do I write this fraction$\frac{1}{{x}^{a}}$as${x}^{n}$What happens to the $a$? I'm confused. wanaopatays 2022-05-22 Answered

### What is the ' factorization of 50? Kendrick Pierce 2022-05-22 Answered

### The equation $d=\frac{|Ax+By+Cz+D|}{\left(\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}\right)}$ gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector? Kaeden Woodard 2022-05-22 Answered

### How many factors does 144 have?

Getting your pre Algebra solved becomes much easier when you have all the answers to your questions by taking a closer look at the various examples dealing with Pre-Algebra subjects. We have intentionally collected the list of pre-algebra equations and various solving equations with decimals problems to help you see various examples that explain how certain solutions are found. Still, if something sounds unclear or you are concerned about some solution that has been provided, approach pre-algebra with reverse engineering approach (going backwards).