For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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2
votes
1answer
45 views

Let p<q both be prime numbers. Prove that log is not rational number

So i was given a question that starts off like this Prove that $\log_q(p)$ is not a rational number. Recall that $\log_y(x)$ for real numbers $x,y>0$ is defined to be the real number $r$ so ...
4
votes
2answers
61 views

Prove: the countable product of regular topological spaces is regular.

Prove: the countable product of regular topological spaces is regular. Label the countable product of $X_i$ as $X$. Given $x \in X$ and $U$ a closed set s.t. $ x \notin U$, let's find disjoint ...
0
votes
1answer
27 views

Computing the gcd of a relatively prime polynomial

I was given a question that starts off like this. Suppose that $a, b \in \mathbb{N}$ and relatively prime. For each of the following, if the answer must be one particular number, then compute it; ...
0
votes
1answer
17 views

Determining cardinality and inverse

Let the function $\chi: P(Z) \to P(Z)$ be defined by $\chi(B) = B^c$ for any $B \in P(Z)$. (In other words, $\chi$ sends a subset $ B \subseteq Z$ to its complement, $B^c$, i.e. the set $Z - B$.) ...
2
votes
2answers
58 views

Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$

This is Velleman's exercise 3.4.13: Prove that $\exists z \in \mathbb R\forall x \in \mathbb R^+[\exists y \in \mathbb R(y-x=y/x) \iff x \neq z]$. I am am stuck on that one. Seems like I am ...
0
votes
3answers
68 views

Prove that the following pairs of sets have equal cardinality:

(b) $\mathbb Z$ and the set $\{x \in \mathbb R \mid (\exists n \in\mathbb Z)(x = 2^n)\}$ (c) $\{0, 1\} \times \mathbb{N}$ and $\mathbb{Z}$ (d) $\{0, 1\} \times \mathbb{N}$ and $\mathbb{N}$ For ...
3
votes
1answer
29 views

Proof of elements in 4 digits palindrom

Can you prove that there are exactly 90 elements in the set of numbers having 4 digits which are palindromes? This is not a tricky question. I am just trying to understand the concept of proofs ...
0
votes
1answer
25 views

In proving existence and uniqueness of ODE, why do we consider rectangular regions instead of circular regions?

I had this question while reading a proof on existence and uniqueness of solution for ODE...example: http://www.math.uiuc.edu/~tyson/existence.pdf In the proof, function $y' = F(x,y)$ is assumed to ...
5
votes
8answers
987 views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
26
votes
5answers
2k views

When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
0
votes
2answers
43 views

Argument for the diameter of these 2 graphs…

I believe G1 has a diameter of 2 & G2 has a diameter of 4. However, is there a formal way to prove / argue for these given diameters? I'd like to see an argument without having to list all the ...
0
votes
1answer
68 views

Infinite heads from Infinite coin tosses?

If I toss a coin an infinite amount of times, can I be sure to get an infinite amount of heads? Is it possible for it to be tails every flip meaning I get no heads at all?
3
votes
3answers
95 views

Prove if $f(a)<g(a)$ and $f(b)>g(b)$, then there exists $c$ such that $g(c)=f(c)$.

First of all, let me write the statement properly: Theorem : Let $f(x)$ and $g(x)$ are continuous on a closed interval $[a,b]$. If $f(a)< g(a)$ and $f(b)>g(b)$, then there exists a $c$ in ...
3
votes
1answer
39 views

Can some help me understand Zeidler's intuitive proof of Brouwer Fixed Point theorem

On pg53, Zeidler gives the Brouwer's Fixed Point Theorem The continuous operator $A: M \to M$ has a fixed point provided $M$ is a compact, convex and nonempty set in a finite dimensional normed ...
1
vote
1answer
34 views

What functional space does $\mathbb{X} = \{0\}$ belong to?

In a lot of proofs regarding spaces, the example $\mathbb{X} = \{0\}$ is given as the trivial case. Why is that $\mathbb{X} = \{0\}$ is a linear/normed/Banach/Hilbert... space when it is essentially ...
2
votes
0answers
76 views

Help to write a proof (category theory diagram)

It is known that $f$, $g$, $h$ are isomorphisms. It is known that $g\circ f = h^{-1}$. I need to write down the proof of the following theorem. I am an amateur mathematician and am not an expert in ...
0
votes
1answer
27 views

Prove equivalence to the Euclidean Parallel Postulate

Show that this statement (P): The opposite sides of a parallelogram are congruent is equivalent to the H.E.P.P (Q): For every line $l$ and every point $p$ not lying on $l$ there is at most ...
1
vote
1answer
37 views

Can someone help me give a proof for this?

I know there are theorems about integrals of odd and even functions, but i kept wondering about integrals that share symmetry around an axis $x=c$. I've been trying to give a proof for this but can't ...
7
votes
1answer
52 views

Prove that if $\mathcal F \subseteq \mathcal G$ then $\bigcap\mathcal G \subseteq\bigcap\mathcal F$

This is Velleman's exercise 3.3.13. Suppose $\mathcal F $ and $\mathcal G$ are families of sets and $\mathcal F \subseteq \mathcal G$. Prove that $\bigcap\mathcal G \subseteq\bigcap\mathcal F$. My ...
1
vote
2answers
34 views

Show that $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$ where $d>c\in\mathbb{R}$

I'm trying to show that if $d-c>0$, then $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$. In the case where $d-c>\sqrt{2}$, we have: $$ \exists q\in\mathbb{Q}:\sqrt{2}>q>0 \implies ...
0
votes
1answer
52 views

Could someone give a detailed (yet elementary) proof for Jensen's inequality?

I want to prove that Suppose there is a function $f:[a,b] \to \mathbb R$, and there are $x_i \in [a,b], w_i \gt 0 $ for $i=1,\dots,n$ such that $\sum_{i=1}^nw_i=1$, then if the function is convex, ...
1
vote
1answer
94 views

Prove that “No one likes Reggae music” is the same as “Everyone does not like Reggae music”.

I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions I know that similar questions have been asked before about ...
1
vote
0answers
15 views

Proof for two monic polynomial gcds, $d$ and $d_0$, if $d|d_0$ and $d_0|d$, then $d=d_0$

This is an extension to this, that is covered in my higher linear algebra course. I know if $d$ and $d_0$, both $\in \mathbb{F}[x]$ are monic and gcds of some polynomials $g$ and $f$ in ...
1
vote
1answer
23 views

Prove that if $\forall A \in \mathcal F (B\subseteq A)$ then $B \subseteq \bigcap \mathcal F $

This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $. My ...
0
votes
1answer
56 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
2
votes
2answers
52 views

Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$

I have the following problem: Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ Where $F_n$ is the $nth$ Fibonacci number. Proof Basis $n = 6$. $F_6 = 8 \geq 2^{0.5 \cdot ...
1
vote
1answer
53 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
1
vote
1answer
75 views

Are ther situations when 3 points do not lie on a circles?

Consider 3 points on a plane, points are real. Is it possible that the points are placed in a way that makes it impossible to draw a circle trough them. I know that if the point forms a line then ...
2
votes
1answer
41 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
0
votes
1answer
28 views

Analytic version of Hahn-Banach using geometric version

When studying the Hahn-Banach theorem, one can demonstrate the geometric version from scratch and use it to prove the analytic version, as is outlined in Hahn-Banach theorem: 2 versions. To do so, it ...
1
vote
2answers
46 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
0
votes
1answer
26 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
3
votes
4answers
122 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
1
vote
3answers
79 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
votes
3answers
76 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
3
votes
4answers
106 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
1
vote
1answer
46 views

Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.

I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through ...
2
votes
3answers
55 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
1
vote
0answers
24 views

How to prove partial ordering formally?

The question is: The set $S$ is defined as $\varnothing \in S$, If $x \in S$, then also $\{x\} \cup x \in S$. Prove or disprove it is partial ordering. So the set $S$ looks ...
2
votes
2answers
32 views

Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
1
vote
2answers
67 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
1
vote
1answer
40 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
0
votes
1answer
21 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
0
votes
2answers
60 views

Velleman's exercise $3.1.7$

Prove that if $a^3>a$ then $a^5>a$. Velleman gives this "hint": $$\text{One approach is to start by completing the following equation:}\ (a^5-a)=(a^3-a) \cdot x$$ I don't understand this ...
1
vote
1answer
50 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
1
vote
2answers
59 views

Can someone explain to me why set proof involve the words “or” and “and”

For example, on proving the distributive law of set theory, the following constitutes as a proof Proof : I am new to proof involving sets but this to me seems nothing more than replacing unions ...
1
vote
4answers
40 views

Geometry question about lines

If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting ...
4
votes
1answer
35 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
2
votes
3answers
32 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
2
votes
2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...