1
vote
1answer
47 views

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 ...
2
votes
2answers
38 views

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 ...
2
votes
4answers
457 views

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 ...
1
vote
4answers
115 views

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, ...
0
votes
1answer
34 views

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 ...
1
vote
1answer
59 views

Proof by contradiction: logarithm

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 ...
0
votes
3answers
41 views

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 ...
0
votes
2answers
61 views

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 ...
0
votes
1answer
33 views

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 ...
2
votes
5answers
67 views

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 ...
0
votes
2answers
24 views

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: ...
1
vote
3answers
93 views

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 ...
3
votes
0answers
34 views

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 ...
0
votes
1answer
41 views

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?
2
votes
1answer
88 views

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 ...
0
votes
2answers
43 views

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.
4
votes
4answers
191 views

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 $m=0$, ...
0
votes
1answer
40 views

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 ...
2
votes
2answers
83 views

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 ? $x=2$ ...
2
votes
1answer
94 views

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.
1
vote
0answers
36 views

Gradient Proof Question

Prove the given formula ($r=||{\textbf{r}}||$ is the length of the position vector field $\textbf{r}(x,y,z)=x\textbf{i}+y\textbf{j}+z\textbf{k}$). $$\nabla \dfrac{1}{r} = \dfrac{-\textbf{r}}{r^3}$$ ...
7
votes
2answers
257 views

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 ...
0
votes
2answers
27 views

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!
0
votes
3answers
27 views

Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
3
votes
2answers
83 views

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 $n$ ...
2
votes
2answers
37 views

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 ...
0
votes
1answer
37 views

Question about the Least squares method

We have $n$ dots: $(x_1,y_1)\cdots (x_n,y_n)$. We know that if we use the Least squares method we will get a line $y=mx+b$ that giving the minimal value for the function $w=\sum_{i=1}^n ...
1
vote
1answer
32 views

Proof: If $F^3 = F$ then F is diagonalisable

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$. Show: F is diagonalisable. $F^3 = F$ is equivalent to $F^3 - F = 0$. Now I know that ...
12
votes
3answers
1k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
0
votes
1answer
11 views

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 ...
0
votes
1answer
35 views

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 ...
0
votes
3answers
46 views

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 ...
0
votes
2answers
28 views

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 ...
3
votes
4answers
56 views

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 ...
1
vote
2answers
36 views

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 ...
0
votes
1answer
17 views

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 ...
4
votes
2answers
81 views

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 ...
1
vote
3answers
62 views

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: ...
0
votes
2answers
56 views

Prove that for every integer $n \ge 1$, $1 + \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+ … +\frac{1}{\sqrt{n}}\le 2\sqrt{n}$

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 = ...
-1
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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} ...
0
votes
1answer
20 views

Trying to show that a vector value function has equal mixed partial derivatives

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 ...