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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0answers
14 views

Proofs regarding about all Second Derivative Test cases (Inconclusive & Single Variable)

This is how I would prove f''(c) > 0 that f(c) has local min and I would easily flip the inequalities and state a conclusion for f''(c) < 0 that f(c) has local max. Quick Proof for f''(c) > 0 ...
1
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1answer
84 views

The fix points of the Möbius transformations are the eigenspace of a certain matrix.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
0
votes
3answers
32 views

Proving quadratic inequalities?

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
0
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1answer
20 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
0
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1answer
21 views

Proving: If a function is bounded, then the fuction's limit is bounded.

The question I have to answer is the following: Let I be an open interval that contains the point c and suppose that f is a function that is defined on I except possibly at the point c. If $m \le ...
-2
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1answer
26 views

Linear Algebra: Diagnolization [closed]

let $T$ be a linear operator over vector space $V$ over the field $F$, and let $g(t)$ be polynomials with coefficients from field $F$. Prove that if $x$ is an eigenvector of $T$ with correspond ...
3
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2answers
63 views

Can the choice of epsilon be arbitrary in epsilon-delta proofs?

I've been reading Spivak's chapter on limits and something that I don't feel I understand entirely is how the epsilon is decided upon. It makes sense to me in the context of $\,|f(x)-L|<\epsilon$ ...
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2answers
26 views

Show that $a c\equiv b c\pmod m,\;a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m$ [closed]

Show that $a c\equiv b c\pmod m$ with $a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m.$
2
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0answers
27 views

Proof without using the proof of contradiction

By using the proof by contradiction I can determine that the root of a prime number is irrational. But how can I proof this by using the rational roots test to find rational factors of $x^n - p$. How ...
0
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1answer
21 views

Strong induction on a sequence, proving two functions are equal?

Excuse the poor title, but my understanding is still a little fuzzy. Admins feel free to change it Here is the question from the book. suppose that $f_{0}, f_{1}, f_{2}...$ is a sequence defined ...
2
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4answers
40 views

Proving binomial coefficients identity [duplicate]

Let $n$ and $r$ be positive integers with $n \ge r$. Prove that: $$\begin{pmatrix}r\\r\end{pmatrix} + \begin{pmatrix}r+1\\r\end{pmatrix} + \dots + \begin{pmatrix}n\\r\end{pmatrix} ...
0
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0answers
27 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
1
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1answer
32 views

Finding all z (complex) that satisfies an equation

I'm having a little trouble with this problem. It's asking to find all $z\in\mathbb C$ that satisfy $z^3 = -2(1+i\sqrt{3})\overline z$, and to keep the answers in standard form. I tried expanding ...
0
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0answers
37 views

Good source for my “language of math” class?

I'm having a hard time in my "language of math" class (proofs, sets, etc). Right now we're doing finite sets. Are there any good online resources for this class? Thanks
0
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1answer
18 views

prove characterstic polynomial of $2\times 2$ matrix is $C_{A}(x)=x^2-(\lambda_{1}+ \lambda_{2})x+\lambda_{1} \lambda_{2}$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $\lambda_{1}, \lambda_{2}$ not necessarily distinct, be the eigenvalues of A. Show that $$ ...
1
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1answer
23 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
1
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0answers
43 views

To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
0
votes
2answers
44 views

How do I prove this using proof by contradiction

There is a set a set $S$ of numbers. i.e. $(s_1, s_2, s_3, s_4, s_5, ..., s_n)$. The average of the numbers in the set is $N$. How do I prove that at least one of the numbers in the set is greater ...
0
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0answers
16 views

Poles of analytic functions are isolated

Can the set of poles of an analytic function $f:G\rightarrow \mathbb{C}$ contain a limit point? I know that the answer is no for open $G$, but after thinking more I have become paranoid about ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
4
votes
1answer
103 views

Product identity for $n^n$

I came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k ...
0
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3answers
21 views

Divisibility induction proof - question about fractions

I have a question about the example of divisibilty induction proof. Here's the problem [the expression must be divisible by 8]: $5^{n+1} + 2*3^n + 1 = 8*k$ I know that probably I have to proceed ...
0
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1answer
13 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...
0
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1answer
18 views

Prove the length of a linear combination of an orthonormal basis

Let $B=\{u_1,u_2,...,u_p\}$ where $B$ is an orthonormal basis for a subspace $W$. Let $v$ be any vector in $W$, where $v=a_1u_1+a_2u_2+...+a_pu_p$. prove that $$||v||^2=a_1^2+a_2^2+...+a_p^2$$ So ...
4
votes
0answers
49 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...
0
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0answers
9 views

Prove the A x B lexicographical ordering is partially ordered

Is this proof? I think I may have the right ideas, but I'm not sure.
0
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0answers
27 views

Function Proofs

Suppose that $f : A → B$ is a function. If $S ⊆ A$, then we define $f(S)$ to be the set $f(S)={f(x) : x∈S}$. (So for example, if $f : R → R$ is given by $f(x) = x^2$, then we have $f({1,2,3}) = ...
0
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1answer
9 views

Polynomial Function

Show that there is a polynomial function $f$ of degree $n$ such that 1) $f'(x)=0$ for precisely $n-1$ numbers; 2) $f'(x)=0$ for exactly $k$ numbers $x$ if $n-k$ is odd. For part 1: I was thinking ...
4
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0answers
50 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv ...
0
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1answer
27 views

Bijection Proof on

Can anyone help me prove this? This is what I have so far.
3
votes
2answers
48 views

Prove that $4^{(p-1)/2}\equiv 1\pmod p$

If $p$ is a prime of the form $4k+3$, prove that $4^{\frac{p-1}{2}}\equiv 1\pmod p$. I was solving a problem and it came down to this. I have no idea how to prove it, I have tried. Any help would be ...
2
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2answers
19 views

Help proving this recurrence relation?

Let $P_n$ be the number of strings of length n formed from letters A, B, C, E, O, that do not contain two consecutive consonants (that is, B or C). For example, AABOCA and BACOOEBO satisfy this ...
0
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1answer
26 views

A question about the elementary symmetric polynomial

I have asked this question and have come up with a possible answer $$ \frac{d^j}{dx^j}[\frac{(x)_c}{j!}] = e_{c-j}(x,x-1, \cdots ,x-c+1) $$ My first question is, how can I prove this? It seems trivial ...
1
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0answers
15 views

The Jugs of Water Problem - with constraints

Given three jugs containing any amount of water such that a1 <= a2 <= a3 and each jug is large enough to contain all the water, show that it's possible (or not) to empty one jug. Only ...
3
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3answers
74 views

Prove that $\int_a^b x^2 dx = \frac{b^3-a^3}{3}$

I cannot assume that the integral exists as this is part of the exercise. I'm only allowed to use the definition of the integral, which is the following: Let $f$ be defined on $[a,b]$. The function ...
0
votes
1answer
21 views

An induction proof in a set.

I have an induction problem that I have no idea how to start. So the question goes like this. Let $x_1=1$, $x_2=2$ and $x_n=x_{n-1} + 2x_{n-2}$. Prove that $x_n=2^{n-1}$ for all $n$ in the natural ...
1
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0answers
34 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
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4answers
74 views

Mathematical Proofs

Let $A,B,C,D$ be any sets. Prove $(A \setminus B) \times (C \setminus D) \subseteq (A \times C) \setminus ( B \times D)$. Show by way of a counterexample that the reverse inclusion is false. What ...
0
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1answer
33 views

Let H be a subgroup of a group G. Prove that the following statements are equivalent.

Let H be a subgroup of a group G. Prove that the following statements are equivalent. (a) For all $a,b \in G, (aH)(bH)$ is a left coset of $H$ in $G$. (b) For all $a,b \in G, (aH)(bH) = (ab)H$. (c) ...
0
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0answers
38 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
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1answer
33 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
2
votes
2answers
76 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
47 views

Proof, that $a \equiv 1 \pmod{p}$

Let $n \in \mathbb{N}^{+} \smallsetminus \{{1}\}$ and $p = min\{p \in \mathbb{P} : p \mid n\}$. Also, let $a \in \mathbb{Z}$ and $a^n \equiv 1 \pmod{n}$ I need to proof, that $a \equiv 1 \pmod{p}$. ...
0
votes
1answer
37 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
1
vote
2answers
30 views

Show a limit of piecewise function does not exist as x tends to 0

So I am given a function, f(x), as follows: f(x)=sin(1/x) if $0<x\leq1$ and f(x)=4 if x=0. clearly this is an example of a function integrable on [0,1] but it is discontinuous at x=0. How ...
0
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1answer
27 views

induction with factorials

I need help with this please. I understand step one is to let $n=1$. step two let $ n = k$. Step three prove for $k+1$. But I would like a clear example of each... Prove $$\sum_{i=1}^n ...
0
votes
1answer
29 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
1
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1answer
68 views

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$

Consider for real $a<b$ and real $A,B,C$, the function $f:[a,b] \to \mathbb R$ defined by $$f(x) = \begin{cases} A & x = a \\ B & x=b \\ C & a < x < b \end{cases}$$ I want to ...
0
votes
1answer
33 views

Prove the infimum and supremum of the positive rational numbers

I am having this set: $$ X= \mathbb{Q^+} = \{x \in \mathbb{R} \ \ |x \in \mathbb{Q} \ \text{and} \ x>0 \} $$ How can I prove that $\inf X= 0$ and there is no supremum ? (I think there is no ...
0
votes
1answer
29 views

Prove by contrapositive: Φ∪{β} ⊨ α & Φ∪{¬β} ⊨ α iff Φ ⊨ α

We are to prove this by contrapositive (by the way: Φ is a set of formulas of predicate logic and α a formula of predicate logic) I've managed the Right to Left proof, but I struggle with the Left to ...