For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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14 views

Are these solutions to Linear Diophantine Equations too? Where'd they hail from?

(1) Can't the signs - I colored them in red - of x and y be switched? Aren't $x = x_0 - bn/d$ and $y = y_0 + an/d$ also solutions? They satisfy $ax + by = c$? (2) How can I remember these ...
0
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3answers
42 views

Abstract Algebra Quotient Group and Isomorphism Proof Help

If $G$ is an abelian group, $S = \{ y \in G \; : \; y = x^2\; \exists x \in G\}$, and $T = \{ a \in G \; :\; a^2 = e\}$, then $G/T$ is isomorphic to $S$. Proof: Let $G$ be an abelian group, $S = \{ ...
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0answers
19 views

Proofs for linear inverse problems

Could you please help me in proving these two questions? 1- Assuming that $A^{T}Ax = A^{T}b, (A^{T}A+F)x^{'} = A^{T}b$, and $2||F||_{2}\leq\sigma_{n}(A)^{2}$. I want to prove that if $r = b-Ax$ and ...
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2answers
45 views

Why does reflecting a point (x,y) about y=x result in point (y,x)?

I noticed that whenever reflecting a point (x,y) about the line y=x the x and y coordinates become swapped in order to give (y,x). However, I do not know why this is the case. Is there any way to ...
2
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2answers
32 views

Suppose F and G are families of sets. Prove that F and G are disjoint iff for all A∈F and B∈G, A and B are disjoint

I am trying to work through this homework problem but I am having trouble getting past how to get started. Could help with setting up this to prove? I know I need to prove by contradiction and ...
6
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2answers
127 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable parametrisations for ...
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1answer
24 views

How can I show this trigonometric identity?

Using only the basic identities ($\sin^2{A}+\cos^2{A}=1$, $1+\cot^2 A=\csc^2{A}$ and $1+\tan^2 A=\sec^2{A}$) show that: $$ \frac{1}{\csc{A}-\cot{A}}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\csc ...
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2answers
52 views

Max of sum $\le$ sum of max simple proof

I'd like to show the following: $\max_x \sum_y f(x,y) \le \sum_y \max_x f(x,y)$ I've also been asked to show the conditions under which the two are equal. I believe that I'm getting confused by the ...
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2answers
50 views

A question on tangent vector fields

I'm currently taking an introductory course in Differential geometry of curves and surfaces. I have a question on vector field, $\textbf{w}$: Given $p \in S, ...
1
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2answers
35 views

Question about something I found in a proof

I came across in a proof I was reading in my textbook that $ab - a'b' = ab - ab' + ab' - a'b'$. I was wondering why/how that equality is true.
1
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1answer
37 views

Proving a Bound for Oddtown-Eventown or Clubtown

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of ...
2
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3answers
46 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
0
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3answers
39 views

proving $\lim(1/y_n)=1/y$

I have a question on the proof of $\lim (1/y_n)=1/y$ under $\lim y_n=y$; $|y_n-y| < \epsilon |M|/|y|$ $y_n$ is convergent so that it's bounded by some number $|M|$; ...
0
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1answer
13 views

Proof with suprema

Given two non-empty upper-bounded sets $A$ and $B$ composed entirely of positive numbers where $\alpha=sup(A)$ and, $\beta=sup(B)$, and $C = \{ab | a \in A, b \in B\}$, prove that $C$ is ...
2
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2answers
158 views

Help with graph induction proof

I'm trying to prove : Given a simple graph G with n vertices, where n is even, prove that if every vertex has degree n/2 + 1, then G must contain a (simple) 3-cycle. A (simple) 3-cycle is a set of 3 ...
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1answer
38 views
+50

Problem using Stokes's Theorem - Boundary Curve, Unit Normal Vector [Stewart P1097 16.8.5]

$\Large{1.}$ How does one determine the boundary curve, denoted as C, to be the plane $z = -1$? I’m flummoxed because $S$ here is given as bottomless. I'm not enquiring about formal or rigorous ...
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1answer
37 views

A question to the proof of a lemma in Enderton's Mathematical Introduction to logic

I'm referring to the proof to Lemma $25\text{B} \ $,pg$\ 133$ of Enderton's Mathematical Introduction to Logic($2^\text{nd}$ edition): $\overline s(u^{x}_{t})=\overline {s(x|\overline s(t))}(u).$ The ...
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3answers
26 views

Help in proving $\lim \limits_{n \to \infty} (a_n \pm b_n) = a \pm b$

This was left as a non-assessed exercise, and I am unsure of how to attack it: If $a_n$ and $b_n$ are convergent sequences converging to $a$ and $b $ respectively. Then prove that: $$\lim \limits_{n ...
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1answer
30 views

Easier/Faster Way to Calculate $\iint \operatorname{curl}\mathbf{G \cdot} \; d\mathbf{S} \quad$?

Is it always correct to rewrite $\iint \operatorname{curl}\mathbf{G \cdot} \; d\mathbf{S}$ as $\iint \color{green}{\operatorname{curl}\mathbf{G} \cdot} \, d\mathbf{S} = \iint ...
0
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3answers
18 views

Prove that a function is a bijection?

I am having trouble with this problem: Prove that the function $f(x)=x^2-2x+3$ with domain $x\in(-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: Basically, I try to use the ...
0
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2answers
37 views

If $f$ is increasing on an open interval and continuous at endpoints, it's increasing on the closed interval.

Prove that if $f$ is increasing on $(a,b)$ and continuous at $a$ and $b$, then $f$ is increasing on $[a,b]$. The question then clarifies: "In particular, if $f$ is continuous on $[a,b]$ and $f'>0$ ...
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3answers
927 views

Fibonacci trick and proving it. [duplicate]

I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is: ...
1
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2answers
23 views

Help with Discrete Math Functions and Bijections

I have trouble with the following problem: Prove that the function $f(x)=x^2-2x+3$, with domain $x\in (-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: I tried to first prove ...
0
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1answer
26 views

Proof for a graph distance

For Graph $G$, there are several $(x, y)$-paths; the shortest among them have length $2$. Thus $d(x, y) = 2$. Prove that graph distance satisfies the triangle inequality. That is, if $x,y,z$ are ...
0
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1answer
32 views

Simple Cycle Graph proof

How can I show/prove that given a simple graph G with $n$ vertices, where $n$ is even, that if every vertex has degree $\frac{n}{2} + 1$, then G must contain a (simple) 3-cycle
0
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0answers
25 views

Linearly Independent and Span Proof

Let R be a field, M be an R-module, $X \subseteq M$. Show that $X$ is linearly independent if and only if $x$ not $\in$ span (X\ {x}) for each $x \in X$ (I'm not sure how to write the symbol for ...
0
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1answer
27 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
0
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1answer
36 views

An n-bit boolean function maps 0/1 strings to 0 or 1

$f: \{0,1\}^n -> \{0,1\}$ The function "depends on i" if there exists two $o/1$ strings (A and B) where A and B differ only at position i and $f(A) \not= f(B)$. How many n-bit Boolean functions ...
0
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0answers
17 views

2 column proof of The Tangent-Chord Angle Corollary

I need to prove 12.23 on this section (http://i.imgur.com/M5iev9K.png) I can use any of the theorems or corollaries before 12.23 but not the ones after it. This is a list ...
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1answer
60 views

Using Pigeonhole Principle for a graph proof

Using the Pigeonhole Principle, prove that in any graph with two or more vertices there must exist two vertices that have the same degree. (Note: the problem does not assume that the graph is ...
0
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0answers
11 views

Help with Integer Modulo Proof

I am stuck on this problem for a while and need some help: Prove that for any prime $p$, if $[a]*[b]=[0]$, does it follow that $[a]=[0]$ or $[b]=[0]$? Work: I do not know where to start. I was ...
0
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1answer
51 views

Prove it is possible to pick 11 integers whose sum is divisible by 11 [closed]

I'm not sure how to write a formal proof for this problem. I can come up with multiple examples, but I need to prove formally that there exists a subset of 11 integers whose sum is divisible by 11. ...
0
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0answers
10 views

Building a nonsingular curve

I have a quintic surface, defined by a homogeneous polynomial in the variables $x,y,w,z$ in $\mathbb{P}^3$. I know the polynomial to have the form $$ xP_1 + wP_2 + (y-z)^2(y^3 + z^3)=0 $$ where the ...
0
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1answer
14 views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
1
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2answers
33 views

Ring with special rules for add and mult

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$. What integer in the ring $R$ is ...
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0answers
24 views

Show that the field Q[sqrt2] cannot be ordered using the defined relation

The complete questions states: On $\mathbb Q\:$[$\sqrt2 $] we define the relation: $\mathbb a+b\sqrt2 < a'+b'\sqrt2$ if $\mathbb a<a'$ and $\mathbb b<b'$ then show that the field $\mathbb ...
0
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1answer
25 views

GCD of polynomials

In $\mathbb{Z}/5\mathbb{Z}[x]$ use the Euclidean Algorithm to find the GCD of $x^4+x^2$ and $x^4+x^3+3x$. My thoughts: I am getting to the point where I need to use a fraction to get further in ...
0
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0answers
15 views

Question regarding a surjective function

The function $f :\mathbb{Z}_{12} \longrightarrow \mathbb{Z}_4$ is defined by $f(x)=3x$. I am asked given any $a,b \in \mathbb{Z}_{12}$ if preservation of addition and multiplication occur. I proved ...
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2answers
52 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
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0answers
23 views

What is the purpose of the Distributive law in this proof? (From Spivak)

So I'm reading Spivak Calculus and it makes sense. But, what I can't understand is, what is the purpose of the following: $$\begin{align}13\cdot4 &= (1\cdot10 + 3)\cdot4\\&= 1\cdot10\cdot4 ...
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0answers
49 views

Modular arithmatic

Suppose that f : Zmod12 -> Zmod4 is defined by f [x] mod 12 = [3x]mod 4 where the subscript indicates the appropriate modular arithmetic. (A) IS f surjective? (B) is f injective? (C) let [a], [b] be ...
0
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1answer
35 views

Symmetric Positive Definite Matrix Proof

Suppose that $H^+ = H - (\mathbf y^TH \mathbf y)^{-1} H\mathbf y \mathbf y^T H + (\mathbf y ^T \mathbf s )^{-1}\mathbf s \mathbf s^T $ where H is symmetric and positive definite. Supposing that ...
0
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0answers
49 views

Let {$p_n$} be a sequence of points in the $\mathbb{R}^2$. Use the notion of convergence to solve the following

A) Define what it means for a point p $\in$ $\mathbb{R}^2$ to be a limit point of {$p_n$}. B) Prove that p is a limit point of {$p_n$} if and only if {$p_n$} has a subsequence which converges to p. ...
1
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4answers
45 views

proof by induction: sum of binomial coefficients

Question: Prove that the sum of the binomial coefficients for the nth power of $(x + y)$ is $2^n$. i.e. the sum of the numbers in the $(n + 1)^{st}$ row of Pascal’s Triangle is $2^n$ i.e. prove ...
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2answers
23 views

Question with this proof

The integer $m$ is odd if and only if there exists q $\in \mathbb{Z}$ such that $m=2q+1$ I know that $m$ is even if 2|n, and $n$ is odd if $n$ is not even. I also know the division algorithm, which ...
1
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1answer
30 views

Proving modulo equation with x-power

I'm trying to prove following equation: $$ (g^{y} \mod n)^x \mod = g^{xy} \mod n $$ I tried many multiple approaches, all of them failed, and there is waaay too much of them to write them here, so I ...
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1answer
40 views

Prove combinatorial identity

Prove the following identity: $$ {{i+j}\choose{i}}\left\{{n}\atop{i+j}\right\} = \sum_{k=0}^n{{n}\choose{k}}\left\{{k}\atop{i}\right\}\left\{{n-k}\atop{j}\right\} $$
1
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2answers
30 views

How to prove that certain points relating to a trapezoid are collinear?

Can you help me to prove that in any trapezoid, which is not a parallelogram, the following points are collinear? The midpoints of its bases. The point of intersection of diagonals. The point of ...
1
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2answers
51 views

Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
1
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2answers
74 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...