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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0
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1answer
21 views

Determining the image of a function [duplicate]

I was given a function that says: What is the image of the function $F: \Bbb Z \times \Bbb N \rightarrow \Bbb R$ given by $f(a,b) = \frac{(a-4)}{7b}$ I need help really understanding how to find an ...
1
vote
2answers
41 views

Calculus Spivak. Chapter 1. Question 1. (i) or are there many ways of skinning a cat

I'm taking on Spivak's Calculus a little later on in life via self-study as i'm looking to improve my CS abilities and have always been interested in Maths but unfortunately didn't have the chance ...
0
votes
2answers
29 views

Prove or disprove the following asymptotic relations

$P(x) = 2^x$ Prove or disprove that $p(n^3 + 4) \in O\left(p\left(n^3\right)\right)$ $2^{(n^3 + 4)} \in O(2^{n^3})$ $\lim_{n \rightarrow \infty} \space \frac{2^{n^3 + 4}}{2^{n^3}}$ using ...
1
vote
2answers
29 views

Reflexive, Symmetric, and Transitive on a relationship defined as “m-n is odd” proof

Main question: Is my solution for this proof correct? Also, I have some questions about my solution and the definitions of Reflexive, Symmetric, and Transitive. Here is the question and here is my ...
-2
votes
1answer
46 views

How do I solve this prove of matrix? [closed]

Let $Ax = 0$ be a homogeneous system of $n$ linear equations in $n$ unknowns that has only the trivial solution. Prove that if $k$ is any positive integer, then the system $A^k x = 0$ also has only ...
0
votes
1answer
35 views

Please help me solve this tautological proof

I'm studying for an upcoming exam and have run across this tautological proof: $(R\to Q)\to ((J\land\neg K)\to [(J\equiv Q)\lor(K\equiv R)])$ To start this one off, I decided to create two ...
0
votes
1answer
41 views

Which of the following are bijections?

• $f : \mathbb{Z} → \mathbb{Z} \\ f(x) = x^5 - 3$ • $g : \mathbb{R} → \mathbb{R} \\ g(x) = x^5 - 3$ • $h : \mathbb{Q} → \mathbb{Q} \\ h(x) = x^5 - 3$ • $F : \mathbb{R} → [0, ∞) \\ F(x) = e^x$ ...
-2
votes
1answer
41 views

Logic: Conditional Proof

$(G\land H)\to (J\equiv L)$ $(G\equiv H)$ $(H\land\neg L)\lor(H\land K)$ | $J\to K$ I am trying to use a conditional proof to solve this one. So I'm assuming J is true and using that to prove ...
0
votes
1answer
34 views

solving for the inductive step in a proof by induction

I have no trouble solving for the base case. I need help solving the inductive step. I know that the nth line creates n new regions. But I don't know if that's based on intuition or if I have to ...
2
votes
2answers
71 views

prove if f(x) has an infinite limit then limit of 1/f(x) is = 0

I wanted to ask if someone can do me the favor pointing out the mistakes I might of made in proving the theorem below. Also is there a way to prove the theorem without using the definition of limits? ...
1
vote
1answer
25 views

Computing the GCD

So I was given multiple questions of computing the GCD of $\gcd(10;45)$ and $\gcd(1701;3768)$, etc. The questions generally worked with numbers and I was able to solve it quite simply since I knew ...
4
votes
2answers
515 views

Proof of a discovery involving the square of whole numbers

It was probably discovered by someone else but: When you take the square of a non-zero whole number the sum of the numbers digit is always equal to $1,4,7,9$ How can I write a mathematical proof of ...
3
votes
4answers
100 views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part ...
2
votes
1answer
48 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
62 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
60 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
69 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
994 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 ...
28
votes
6answers
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
46 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
71 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
100 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
28 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
38 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
96 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
57 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
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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
76 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
124 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
108 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 ...