2
votes
2answers
36 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 ...
0
votes
1answer
18 views

Integrability condition on the Fourier transform

I have a question for This is much healthier who answer my query- This is a problem from the book from Stephane Mallat "A wavelet Tour of signal processing: a sparse way". A function $f$ is bounded ...
1
vote
1answer
53 views

My proof is wrong, can anyone tell me why?

$$\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}, [x(x+1) = y(y+1)] \Leftrightarrow [x = y]$$ $$\forall x \in \mathbb{Z} , \forall y \in \mathbb{Z}, [x(x+1)=y(y+1)]\Leftrightarrow [x=y]$$ ...
0
votes
1answer
43 views

Trigonometry - proving an inequality

I came across this question while doing trigonometry. I have tried everything that I could possibly think of, AM/GM, converting it into quadratic equation, conditional identities, solving from RHS, ...
1
vote
1answer
23 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
1
vote
1answer
36 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
0
votes
2answers
47 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
1
vote
3answers
47 views

Proof for modulus via direct or contrapositive

I have to prove the following via direct proof or via contra positive. For $a,b\in \mathbb{Z} $ it follows that $ (a+b)^3 \equiv a^3 + b^3 \mod 3$ I'm unsure of the best way to approach this ...
1
vote
1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
1
vote
3answers
30 views

Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
1
vote
1answer
31 views

Correctness of Proof by Refutation

I am trying to solve the following by proof by refutation: A or B 1. NOT A or I 2. NOT B or T 3. NOT I 4. NOT T 5. Where the goal is to prove a contradiction. ...
4
votes
1answer
151 views

How to prove that equilateral triangle formed by cube's corners cannot be fully inserted to this cube

I would like to prove that equilateral triangle prescribed by cube's corners and sides equal to $b = a \sqrt{2}$ cannot be inserted into the interior of a cube of side $a$. This triangle is presented ...
2
votes
1answer
73 views

Prove that the dual graph of any (planar) graph is connected

I'd like to know if there's a standard proof that the dual graph of any planar graph is connected (or, if there's a counterexample, I'd like to know that too). I've thought of a proof that might work ...
2
votes
0answers
32 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
1
vote
3answers
28 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
3answers
47 views

Disproving 0 as a dividend

Prove each of the following statements. (a) For all $b \in \mathbb{Z}$ if for all $k \in \mathbb{N}$, $b \not\mid k$, then $b = 0$. By hypothesis: $b \not\mid k \implies b\ell \neq k, \ell ...
2
votes
1answer
54 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
0
votes
1answer
69 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
vote
1answer
14 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
votes
1answer
20 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
1
vote
1answer
40 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
0
votes
1answer
32 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
0
votes
1answer
30 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
2
votes
2answers
293 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
1
vote
2answers
22 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
2
votes
2answers
21 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
0
votes
0answers
39 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
0
votes
1answer
33 views

Are these proofs correct?

I haven't formally learn how to do proofs, but I attempted some of these. It'd be great if you guys can check them and give me some pointers. Thanks!
1
vote
2answers
60 views

Ground Plan - Prove Fermat-Euclid's Totient Theorem with Lagrange's Theorem

If $\gcd(a,n) = 1$, then $a^{\phi(n)}\equiv 1\pmod n$. Here's a three-step proof. An integer a is invertible means there's some $a^{-1}$ such that $aa^{-1}\equiv 1 \pmod n$. By cause of Jones p84 ...
4
votes
2answers
166 views

Fermat's Little Theorem fails for composite instead of prime numbers.

I know Fermat's Little Theorem = Fermat-Euler's Totient Theorem when $n$ is prime. Elementary Number Theory, Jones, p83 writes if we simply replace p with a composite integer n, then the ...
2
votes
1answer
59 views

Backward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

(1) How can you preconceive to prove by contradiction? Prove by contradiction. Suppose $n$ is composite. This means there exists a divisor $d|n$ such that $1<d<n$. We are given that ...
1
vote
2answers
38 views

Ground Plan – Forward direction – Wilson’s Theorem – p is prime $\iff (p-1)!\equiv-1(mod\ p) $.

Lemma 5.3 - I omit proof here - Let p be prime. Then $x^2 \equiv 1 \, (mod p) \iff x \equiv \pm 1 \; (mod p)$ First we establish the result for the first two primes 2, 3. Then prove the result for ...
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) ...
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 ...
2
votes
1answer
42 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
3
votes
1answer
75 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
0
votes
1answer
41 views

Direct Product of Rings and Atoms

Which elements in a direct product $\prod _{\lambda \in \Lambda} R_\lambda$ of rings are atoms? Prove your answer. (Assume this is a commutative ring) My Answer: Atoms of $\prod R_\lambda$ are ...
1
vote
2answers
105 views

Cauchy Sequences are Bounded. Questions on proof (Abbott p 59 lemma 2.6.3)

By the agnecy of p 44 Definition 2.3.1, we find M > 0 such that $|a_n| \le M$ for all $n \in N$. Not duplicate. Proof. By definition, given any $e >0$ there is an integer $N$ such that $|a_{n}- ...
2
votes
2answers
53 views

Contraction Mapping Theorem. Any $\{ x,f(x),f(f(x)),\ \ldots) \} $ converges to the unique fixed point of f. (Abbott p 114 q4.3.9 d)

Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$. Parts a,b. (c) Prove that $y$ is a fixed ...
2
votes
1answer
30 views

Proof confirmation: If $\gcd(u,v)=1$ and $uv$ is a square, then $u$ and $v$ are squares.

This is a problem from my workbook(not homework), and I can tell that it is true simply upon observation(They share no factors[other than one] and they, when multiplied have all squares as their ...
1
vote
2answers
72 views

if $g$ is continuous at $c$ and $g(c)\neq 0$, there exists an open interval containing $c$ on which $f(x)/g(x)$ is defined (Abbott p 113 q4.3.5)

Theorem 4.3.4.(iv) says that $f(x)/g(x)$ is continuous at $c$ if both $f$ and $g$ are, provided that the quotient is defined. Show that if $g$ is continuous at $c$ and $g(c)\neq 0$, then there exists ...
6
votes
1answer
91 views

Help with a Binomial Identity: $\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = 2^{n+1}$

The following is a problem from the 5th edition of Niven's An Introduction to the Theory of Numbers: Problem 23 of Section 1.4 asks us to prove that $$\sum_{k=0}^{\infty} \binom{n+k}{k}2^{-k} = ...
1
vote
1answer
90 views

Permutation proofs

I have just started going though "An Introduction to the theory of groups" by J.J Rotman. I have questions above the following two exercises: " The identity function $1_X$ on a the set $X$ is a ...
0
votes
3answers
59 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
3
votes
1answer
292 views

If every convergent subsequence converges to $a$, then so does the original bounded sequence (Abbott p 58 q2.5.4 and q2.5.3b)

Assume $(a_{n})$ is a bounded sequence such that every convergent subsequence of $(a_{n})$ converges to the same limit $a\in \mathbb{R}$. Show $(a_{n})$ must converge to $a$. Prove by contradiction. ...
3
votes
2answers
48 views

A subsequence of a convergent sequence converges to the same limit. Questions on proof. (Abbott p 57 2.5.1)

Solutions to Homework 3 doesn`t duplicate. We have to prove that if $(a_{n})$ is a sequence in $\mathbb{R}$ with $\displaystyle \lim_{n\rightarrow\infty} a_n =a$, and if $(a_{n_{k}})_{k\in ...
2
votes
1answer
43 views

Prove shuffled sequence $\{x_i, y_i\}$ converges $\iff \lim x_n = \lim y_n$ (Abbott p 49 q2.3.5)

Let $(x_{n})$ and $(y_{n})$ be given. Define $(z_{n})$ to be the shuffled sequence $(x_{1}.y_{1},\ x_{2},\ y_{2},\ x_{3},\ldots,x_{n}, y_{n},\ldots)$ . Prove that $(z_{n})$ is conv ergent $\iff ...
1
vote
1answer
63 views

Find an ordered basis of $V$ such that $[T]_\beta$ is a diagonal matrix.

The entire problem statement is: Let $V$ be a finite dimensional vector space and $T:V\to V$ be the projection of $W$ along $W'$, where $W$ and $W'$ are subspaces of $V$. Find an ordered basis ...