For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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6
votes
3answers
132 views

a problem with a false proof

Whats wrong with this: $$\large 1^0=1^2$$ Since bases are same, therefore $$\large 0=2$$ My thinking: Since the function $f(x)=1^x$ is not one to one, therefore whenever $f(x)=f(y)$ $x$ need not ...
1
vote
1answer
18 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
0
votes
0answers
11 views

Orthonormality and fourier transform

If $g\in\mathcal{L}^2(\mathbb{R})$ then $\sum_{k\in\mathbb{Z}} |\hat{g}(\zeta+2k\pi)|^2=1$ for a.e $\zeta\in \mathbb{R} \Rightarrow \{g(.-k): k\in \mathbb{Z}\}$ is an orthonormal system. Please ...
-8
votes
0answers
51 views

Fermat's Last Theorem 1 Page Proof [on hold]

Fermat’s Last Theorem This theorem basically states that $A^n + B^n \neq C^n$, $n \gt 2$ if $A, B, C$ and $n$ are all positive integers. This inequation can be rewritten as $C^n – B^n \neq A^n$. ...
2
votes
2answers
64 views

A problem on nested radicals

Find the value of $x$ for all $a>b^2$ if: $$\large x=\sqrt{a-b\sqrt{a+b\sqrt{a-b{\sqrt{a+b.......}}}}}$$ My attempt $$\large x=\sqrt{a-b\sqrt{(a+b)x}}$$ $$\large x^4=(a-b)^2(a+b)x$$ $$\large ...
2
votes
2answers
55 views

A problem on continued fractions

Find the value of $x$, if: $$\large 1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}$$ My attempt: Noting that: $$\large x=1+\frac{1}{2+\frac{1}{x}}$$ $$x=\frac{1+\sqrt{3}}{2}$$ question: Is my solution ...
1
vote
1answer
36 views

Proving some properties of $\Bbb N$ without using recursion

I will try to prove that if $a, b, c \in \Bbb N$ and $a \in b \in c$, then $a \in c$ (the transitivity property). I will not use recursion or the replacement axiom. Actually we can proove in the same ...
1
vote
2answers
80 views

Prove: If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Can someone tell me if this proof is acceptable? Assume $A \not\subseteq C$. This means that there is an $x \in A$ such that $x \not\in C$. But since $\forall x \in A: x \in B$ and $\forall x \in B: ...
2
votes
1answer
34 views

Is this a valid natural deduction?

I'm trying to prove that $\{(p_1\implies p_2),p_1,(p_2\implies p_3)\}\vdash (p_3\vee p_5)$ which seems easy, but I'm unsure about a step in the way. I did: $1.\ p_1\implies p_2 \text{ (Pre)}\\2.\ ...
1
vote
1answer
33 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
1
vote
1answer
88 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
0
votes
2answers
22 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
2
votes
1answer
47 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
6
votes
1answer
48 views

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set.

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set. This is the result from question part (a). Now for part (b), will it also hold when the ...
0
votes
2answers
91 views

Planar graphs where every face boundary is a cycle of even length are bipartite

Let $G$ be a connected planar graph with a planar embedding where every face boundary is a cycle of even length. Prove that $G$ is bipartite. If every face boundary is a cycle of even length, ...
0
votes
1answer
38 views

If $x$ is an integer and $x \ge 5$, then there's $y$ such that $x + y$ is a perfect square with $x > y$.

$y < 5 \le x$ by hypothesis. Let $y = -x$. Then, $-x + x = 0$. Since $0$ is a perfect square, we are done. I am not sure if this proof would fly. Please, tell me what you think.
1
vote
2answers
41 views

Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the ...
0
votes
1answer
25 views

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism Attempt: Let $\Phi: Z_m \rightarrow Z_n$ be a ring homomorphism ...
1
vote
2answers
31 views

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. My Attempt Shown

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. Attempt: Let $F'$ be the field of Quotients of the field $F$. Let $\Phi:F \rightarrow F'$ such that ...
5
votes
2answers
135 views

Is my proof that $\lim\limits_{n\to +\infty}\dfrac{u_{n+1}}{u_n}=1$ correct?

I'm doing an exercise where $(u_n)$ is a numerical sequence which is decreasing and strictily positive.While $(u_n)$ is a numerical sequence which is decreasing and strictily positive, then $(u_n)$ is ...
0
votes
1answer
22 views

Let $H$ be a simple graph on $n$ vertices that has $m$ edges. Prove that $H$ contains at least $m-n+1$ cycles.

Can someone please verify the proof I just wrote, or offer suggestions for improvement? Also, how do I prove the base case? Let $H$ be a simple graph on $n$ vertices that has $m$ edges. Prove that ...
2
votes
0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
0
votes
0answers
18 views

Number solutions of congruence

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ and $f(x)\equiv 0 \pmod p$ have more than $n$ solutions. Then $p\mid a_i$ for $i=0,1,\dots,n$. My proof: Let $m$ be the maximal such that $p\nmid a_m$ ...
0
votes
0answers
14 views

Prove that both $R_p$ and $R^p$ are abelian groups under ordinary addition of rationals.

Q: Let $p$ be a fixed prime. Let $R_p$ be the set of all those rational numbers whose denominator is relatively prime to $p$. Let $R^p$ be the set of rationals whose denominator is a power of $p\ ...
2
votes
4answers
87 views

Prove $A \subset \emptyset \iff A = \emptyset$

How does one prove this? Can one prove by contradiction by saying: Let $A$ be any set such that $A$ contains at least one element. Now assume $A \subset \emptyset$. This is clearly absurd by the ...
0
votes
3answers
73 views

How many triangles in the figure [on hold]

I got $36$ so far. But I suspect that there might be more of them. Can anyone find more than $35$?
1
vote
1answer
71 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
7
votes
1answer
117 views

Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
0
votes
2answers
51 views

If a converse of an implication is false, does this mean that the proof of that implication will always have an implication that is not reversible?

Let $f:X \rightarrow Y$ be a function and $B_1, B_2 \in \mathcal{P}(Y)$. Prove that $B_1 \subseteq B_2 \Rightarrow \overleftarrow{f}(B_1) \subseteq\overleftarrow{f}(B_2)$. My attempt: $\begin{align} ...
1
vote
1answer
36 views

Find the Green's function for an ODE

$$ Lu= (x-2)u''+(1-x)u'+u , \, u'(0)=(1)=0$$ It can be shown that ${x-1,e^{x}}$ is a fundamental set for $L$ on this interval. $$ g(x,y)= c_1(x-1)+c_2e^{x}, 0\leq x<y , c_3(x-1)+c_4e^{x}, ...
1
vote
0answers
25 views

Use a fixed point argument to show there exists a unique solution to the following BVP

Show using a fixed point argument that there exists a unique solution $f\in C[0,1]$ to $$ -f''(x)+\sin(f(x))=\sin(x) , x\in (0,1), y(0)=y'(1)=0 $$ This is what I have so far: We can show ...
2
votes
1answer
18 views

Showing the slope of a line is a rational number with denominator $2$.

I am looking at Stark's 'An Introduction to Number theory' book, and I'm trying to do the following question from the exercises 7.1: $Q4$: Show that if there are two different integral points with ...
2
votes
2answers
30 views

Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $ I have some ...
0
votes
1answer
58 views

Partial derivative and derivative.

I want to show that if $f:\mathbb{R}^n\to \mathbb{R}$ and $df_a$ is the derivative of the function at $a$ then $df_a(v)=\displaystyle\frac{\partial f}{\partial v}(a)$. I saw a few proofs of this ...
1
vote
1answer
31 views

Given Tf(x), find the equivalent operator m(k)f^(k) in the Fourier transform sense.

Let $f\in L^2(\mathbb R)$, let $$ g(x) = Tf(x) = \int^{x+1}_{x} f(s)ds $$ Find $m\in L^\infty(\mathbb R)$ such that $\hat g(k)=m(k) \hat f(k)$. Use this to show that $T$ is a bounded linear operator ...
2
votes
1answer
51 views

Understanding a proof of Lagrange's four-square theorem

I've been looking at Wikipedia's proof of the four-square theorem and trying to work out the details - I like that it doesn't need to separate the cases for $m$ even and odd, but there is one step ...
0
votes
1answer
80 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
1
vote
1answer
36 views

Proving the Well-Ordering Property

My textbook states the Well-Ordering property as following: If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \le a$, for all $a$ in $A$ ($m$ is called ...
2
votes
2answers
25 views

Recurrent point: two definitions

Let $X$ be a topological space and $T:X\longrightarrow X$ a function. Now lets look at the two following definitions: $x\in X$ is a recurrent point if for every neighborhood $U$ (of $x$) the ...
3
votes
2answers
39 views

Proving $v$, $T(v)$, $T^2(v)$ is a basis

I'm trying to prove the following: Given that $V$ is a vector space, with $dim V = 3$, and $T: V \to V$ is a linear map with the properties $T^2(v) \neq 0$ and $T^3(v) = 0$, with $v \in V$, show that ...
1
vote
0answers
34 views

Can you check my proof concerning an invariant subspace under a diagonilizable linear operator and its complementary invariant subspace?

This was an exercise problem from H&K Linear Algebra(sec 7.2, exercise 18). Could you check my proof? The theorem is as follows: If $V$ is a finite-dimensional vector space and $W$ is an ...
1
vote
0answers
57 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
0
votes
1answer
18 views

A question regarding cyclic quadrilateral

A cyclic quadrilateral $ABCD$ has sides of magnitude $1,2,3,4$ respectively. One of its diagonals is $2.5$. Find the magnitude of the other diagonal. This is how I tried to solve it: For a cyclic ...
1
vote
1answer
52 views

Artin Algebra 2.8.3 “Does every group whose order is a power of a prime $p$ contains an element of order $p$?”

I'm not sure whether or not my answer and proof for this question are valid. Could you point out any flaw? Let $G$ be an arbitrary group, an arbitrary element of $G$ be $g$ and $|G|=p^n$. Since a ...
0
votes
1answer
62 views

Term by term integration

Let $D \subset \mathbb{R}^{d} $ be open. For $u,v \in C_{0}^{\infty}(D)$, we define \begin{eqnarray*} \mathcal{A}(u,v)=\sum_{i,j=1}^{d} \int_{D} \frac{\partial u(x) }{\partial x_{i}}\frac{\partial ...
0
votes
1answer
57 views

am i cheating in this number theory proof?

the question (from burton's elementary number theory); $verify\ that\ \forall n\ge 1,$ $$2\cdot6\cdot10\cdots(4n-2)=\frac{(2n)!}{n!}$$ my work/proof; this is obviously true for $n=1$, so assume ...
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 ...
0
votes
0answers
31 views

Trying to prove that two angles are congruent in a isosceles trapezoid

I was given this assignment to do the following. Write a paragraph proof for the following scenario. Given: KLMN is an isosceles trapezoid. Prove: ∠LKM is congruent to ∠MNL The thing is that I ...
1
vote
0answers
49 views

Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
1
vote
0answers
42 views

Is this proof for the undecidability of $\beta$-normalisation in $\lambda$-calculus valid?

The proofs I have so far seen for the undecidability of $\beta$-normalisation all make use of Gödel numbering in order to first prove the more general Scott-Curry theorem. As an exercise, I have tried ...