# Tagged Questions

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### Is this sufficient for linear independence proofs??

I've been doing all of these proofs the same basically, I just want to make sure I'm doing them right, I didn't include all the details but I have the outlines of my proofs here. 1) U and W are ...
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### Proof About Division of Integers

Here is a problem I just finished working on: Prove that if $n$ is composite then there are integers $a$ and $b$ such that $n$ divides $ab$ but not $n$ does not divide either $a$ or $b$. One ...
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### Proving 7n+5 is never a cubic number?

This is from a question that starts with: An arithmetic progression of integers an is one in which $a_n=a_0+nd$, where $a_0$ and $d$ are integers and n takes successive values $0, 1, 2, \cdots$ Prove ...
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### Proof that arithmetic mean is greater than geometric mean? [duplicate]

I have to prove that $\frac{x + y}{2}> \sqrt{xy}$ algebraically for any $x,y \in \mathbb{R}$ such that $x,y \ge 0$ and $x\ne y.$ I'm fairly confused as to how to solve this problem algebraically, ...
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### Proof by induction and inequalities

I am stuck on this question: given $a_1a_2≤(\frac{a_1+a_2}{2})^2$ prove by induction of m that $$a_1a_2...a_p≤(\frac{a_1+a_2+...+a_p}{p})^p$$ where $a_i$ are all positive and real and $p=2^m$ (an ...
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I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
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### Proof by contradiction proving both numbers are not odd.

I have to do a proof by contradiction: Suppose $a,b,\in\mathbb{Z}$. If $4| (a^2 + b^2)$ then a and b are not both odd. So far I know that I need to prove that if $4|(a^2+b^2)$ then a and b are both ...
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### Proof by contradiction for a set question

I have a statement I need to prove by contradiction: If A and B are sets then A intersect (B-A) = {} (empty set). None of the questions I've ever done for this class are like this so im not really ...
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### Proof regarding division with a remainder

Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$ I've done some of these questions before with modulus and division, but I'm unsure of how ...
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### Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
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### Modulus related proof help

I need to prove this via either direct proof, or contrapositive. Unsure of the best way to approach this. if $a \equiv b\mod n$ and $c \equiv d\mod n$, then $ac \equiv bd\mod n$ So far I have: ...
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### Prove: $\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$

Is there a quick, fancy, way of proving sums such as this? Prove that: $$\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$$ A recent homework assignment I turned in had a couple problems similar to the ...
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### Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
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### Prove in complexity theory

Given a language A, which is in NP and also not NP-complete, I have to prove that P != NP. [Note: A is not trivial] Any suggestions?
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### Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
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### Prove a formula about binoms

I want to prove that $\binom{n}{n/2} \leq 2^{n-1}$ [Assuming $n$ is even] I've tried to do that but I didn't succeed.
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### Prove that p has m distinct roots if and only if p and p' have no roots in common

Problem: Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p'$ have no roots in common. My proof so far: If $m=0$, ...
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### show analytic function such $f(z)={\operatorname{Log}(z+5)\over z^2+3z+2}$

Show that $f(z)=\dfrac{\operatorname{Log}(z+5)}{z^2+3z+2}$ is analytic everywhere except at the point $-1,-2$ and on the ray $\{(x,y):x\le -5,y=0\}$. i think that separate denominator and ...
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### Proving a math statement is true or false

For all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$ Is my logic correct in proving that statement is true ? Can provide an explanation of how to test this proof ? $x=2$ ...
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### Curl Proof Question

Prove the given formula. So far I have $f\textbf{F}=(f\textbf{F}_1, f\textbf{F}_2, f\textbf{F}_3)$, but I'm not sure where to go from there. Could anyone give me some pointers? Thank you.
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Prove the given formula ($r=||{\textbf{r}}||$ is the length of the position vector field $\textbf{r}(x,y,z)=x\textbf{i}+y\textbf{j}+z\textbf{k}$). $$\nabla \dfrac{1}{r} = \dfrac{-\textbf{r}}{r^3}$$ ...
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### An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...
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### How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
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### Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
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### Proof of $\phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd$

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
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### How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
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### Let Cn be the largest possible number of intersection points of a family of $n$ lines in the plane. Prove that $Cn = n(n-1)/2$

(If some lines are parallel, or if three lines intersect at a single point, then the number of intersection points could be less than $Cn$.) Question for proofs homework, which will be on the ...
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### Under what conditions on $a,b$ is $1/(a+bi)=(1/a)+(i/b)$?

Question in proofs review in the complex numbers unit. I expressed $1/(a+bi) = (a-bi)/(a^2+b^2)$ I then separated the two terms in the denominator to get $a/(a^2+b^2)-bi/(a^2+b^2)$ I then equated ...
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### Define a relation on $Z$ by a~b if and only if $a=b(mod2)$ and $a=b(mod5)$. Show that ~ is an equivalence relation.

The if and only if is throwing me off. Would the first direction be to prove the two modular conditions hold if the relation is an equivalence relation? Furthermore, I'm having difficulty proving ...
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### Let A, B and C be sets. Prove that $A \cap (B-C) = (A \cap B) - (A \cap C)$

Someone please edit so the & symbol is the intersect (reverse of U). This is a recent question on proofs homework. From what I understand, intersect and minus symbols used in equations for sets ...
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### Solve $85x \equiv 34 \pmod{153}$

I'm not exactly sure how to solve these modular problems involving a variable. Can someone solve this (trivial) example with explanation? I found the answer (4) by trial and error, however, I'm sure ...
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### Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = hcf(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.

Proofs homework question. We know that if $a|c$ and $b|c$ then $a\cdot b\cdot s=c$ (for some positive integer $s$). $(ab|c)$ Then doesn't $ab|dc$ since $ab|c$? I feel like I'm misunderstanding my ...
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### Let $a$ and $b$ be coprime positive integers. Prove that, for any integer $n$, there exist integers $s$ and $t$ such that $sa + tb = n$

I always sort of took this fact for (well..) fact. Can someone help me with the proof? Does this question have something to do with modulus? Since $a$ and $b$ are coprime ($gcd$ = 1), multiplying ...
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### Prove that if $a$ is a rational number and $a^2$ is an integer then $a$ is an integer.

Question on a proof's review: Proof by contradiction: Suppose $a$ is not an integer. Then $a=p/q$ where $p$ and $q$ are coprime, $q$ is not 0, and $q$ is not 1. Then $a^2 = p^2/q^2$. This is ...
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### Help: Proof via Induction homework problem.

(b) Prove that for every integer $n \ge 1$, $$1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$$ This is the second part of a two part question. Part (a) was the following: ...
I understand that this is an induction question. I start with the base case (n=1): $$1 < 2 \tag{That works!}$$ Induction step: Assume the statement works for all $n = k$, Prove for all $n = ... 1answer 45 views ### Countability of Different Sets [duplicate] (a) Prove that$N \times N$is a countable set (b) Let T be the set of two element subsets of N. Prove that T is countable. This is a question in my exam review package. I missed the lesson on ... 1answer 44 views ###$2^n+1 =xy \implies (2^a|(x-1) \iff 2^a|(y-1))$I'd like my proof to be verified of the following exercise from Niven's The Theory of Numbers. Section 1.1 Problem 52: Suppose$2^n+1=xy$, where$x$and$y$are integers$>1$and$n>0$. Show ... 0answers 47 views ### Proof: There are infinite prime numbers of the form 4k+3 [duplicate] I have to proof if true or wrong: There are infinite prime numbers of the form 4k+3. I want to proof: Yes, this is true. My ideas: 1) Assume - as a contradiction - that there are only infinite prime ... 3answers 69 views ### How can I prove this statement by proving its contra-positive? Prove the following statement by proving its contra-positive: If$ r $is irrational, then$ r^{1/5} $is irrational. I am totally confused! (1) How does proving the contra-positive prove ... 6answers 115 views ### Prove that if$n$is a positive integer then$\sqrt{n}+ \sqrt{2}$is irrational. The sum of a rational and irrational number is always irrational, that much I know - thus, if n is a perfect square, we are finished. However, is it not possible that the sum of two irrational numbers ... 1answer 67 views ### How many integers between 1 and 10,000 are neither squares nor cubes? [duplicate] Question in proofs class. No idea how to figure this one out. I've been giving all questions an honest effort - this seems unorthodox. Please help get me started 4answers 97 views ### If$f \colon A \to B$,$g :\colon B \to C$and$g\circ f \colon A \to C$are bijections. Prove that$f $is 1-1,$g$is onto. From what I understand, one-to-oneness means every element in$A$is mapped to a unique element in$B$. To be onto, means for every$y$in$B$, there exist at least one$x$in$A$from which it can ... 3answers 65 views ### Prove that$A \cup B = A$if and only if$B$is a subset of$A$If$A \cup B = A$then$A$is a subset of$A$and$B$is a subset of$A$. Thus$A \cup B = A$. If$B$is a subset of$A$then it follows that$A \cup B$is a subset of$A$. My solution. It seems ... 2answers 41 views ### Let$a, b, c, m, n$be integers,$m, n$not both$0$. (a) Prove that if$am + bn = c$, then$hcf(m,n)|c$(b) Prove that if$am + bn = 1$, then$hcf(m,(n) = 1$(c) Prove that$m/hcf(m,n)$and$n/hcf(m,n)$are coprime. Question on recent review homework ... 1answer 38 views ### Permutation cycles My tasks are the following : Task 1 : Prove that$ \begin{pmatrix} 1 & 2 & \cdots & r-1 & r \end{pmatrix} = \begin{pmatrix} 2 & 3& \cdots & r & 1 \end{pmatrix} ...
Let $f: R^n \to R^n$. $||x||$ is Euclidean norm. Define $f(x) = xg(||x||)$. where $g: [0, \infty) \to R^n$ is differentiable on $(0, \infty)$. $g$ is not constant. I want to show that every \$i \neq ...