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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1
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1answer
61 views

Is this a valid proof of this math challenge problem?

From a fixed point P not in a given plane, three mutually perpendicular line segments are drawn terminating in the plane. Let a, b, c denote the lengths of the three segments. Show that ...
0
votes
0answers
5 views

Prove of negative transitive relation

Prove that a binary relation R on X is negatively transitive if and only if for each x, z∈X, xRz implies that ∀y∈X, xRy or yRz.
2
votes
0answers
49 views

What's wrong with this proof of Schröder-Bernstein theorem?

In V. A. Zorich's Mathematical Analysis I there is an exercise to Analyze the following proof of the Schröder-Bernstein theorem: $(\operatorname{card} X \leq \operatorname{card} Y) \land ...
3
votes
2answers
21 views

$X \cap (Y \setminus Z) = (X \cap Y) \setminus (X \cap Z)$

As the title suggests, what is the easiest way to see that$$X \cap (Y \setminus Z) = (X \cap Y) \setminus (X \cap Z)?$$
3
votes
1answer
14 views

Indicator Function Distributive Property Proof

This is my first post(: I'm trying to understand how to prove the distributive property using the indicator function. I have made the truth tables and understand how this is proved using set ...
5
votes
3answers
87 views

Infinite set always has a countably infinite subset

I'm trying to show that one infinite always has a countably infinite subset, but I'm confused with something on the proof. Let $S$ be one infinite set. In that case, to show it has one countably ...
22
votes
8answers
3k views

LaTeX/TeX Vs. Mathematica for Typesetting [closed]

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regard to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that I ...
12
votes
5answers
239 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
0
votes
1answer
52 views

Total boundedness for a non standard metric on $\mathbb{R}^n$

I want to prove this two things: 1) $(\mathbb{R},d_B)$ is not totally bounded. where $d_B=\frac{|x-y|}{1+|x-y|}$ and $d_E$ is the Euclidean metric. 2) $B_M(0)$ is totally bounded in ...
5
votes
1answer
42 views

Stating the induction hypothesis

I would like to ask about the best way to state the induction hypothesis in a proof by induction. Just to use a concrete example, suppose I wanted to prove that $n!\ge 2^{n-1}$ for every positive ...
3
votes
3answers
52 views

If a set is countable and infinite, there is a bijection between the set and $\mathbb{N}$

I'm trying to show that if a set $S$ is infinite and countable then there is a bijection $\varphi : S\to \mathbb{N}$. Since $S$ is countable, we know that there is an injection $f: S\to \mathbb{N}$. ...
1
vote
0answers
22 views

problem in understanding some part of proof of theorem

In proof of theorem: for every finite non abelian p-group $G$ of class 2 there is utter automorphism of calss 2 fixing $\Phi(G)$ by counterexample,If we know 1. $G'=\langle[a,b]\rangle$ 2. $H=\langle ...
1
vote
3answers
97 views

Let $a,b,c \in \mathbb{R^+}$, does this inequality holds $\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$?

Does the following statement/inequality holds for $a,b,c \in \mathbb{R^+}$? $$\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$$ I've been thinking for hours and I ...
3
votes
0answers
21 views

Why calculating XOR of consecutive values can be simplified?

I was trying to calculate integer xor of 0..n. I named the function xored(n). Note that in examples below ^ does not mean power but integer xor (like in C or Java language) So, xored(0) = 0, ...
1
vote
4answers
67 views

How to prove that the following sequence will never contains number greater than 3

You may now the following sequence: 1 11 21 1211 111221 312211 13112221 Explanation of the sequence (I've put an hint just for the ones who want to search a bit ...
0
votes
2answers
97 views

How to go about proving that $\cos(\frac{\pi}{2}-x) = \sin(x)$?

I have very little experience writing proofs so I don't know how to begin. I recognize that the statement is always true, but I can't go about proving it without using circular reasoning. How could ...
0
votes
0answers
22 views

Let $R$ be a relation on $A$ and let $S$ be the transitive closure of $R$. Prove that $\text{Dom}(S) = \text{Dom}(R)$.

This is from "How To Prove It". The full exercise also asks to prove that $\text{Ran}(S) = \text{Ran}(R)$ but I was set from the outset on proving that $\text{Dom}(S) = \text{Dom}(R)$ first. Since the ...
4
votes
1answer
705 views

If $\lim f(x) = 0,$ then $\lim 1/|f(x)| = \infty.$

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...
1
vote
0answers
32 views

Basic Set-Theoretic Properties from Halmos

I've been backtracking lately to make sure that I have a solid set-theoretic background before taking measure theory this fall. Here's a few facts I've come across today, and my attempted proofs. Let ...
4
votes
4answers
82 views

Understanding how to use $\epsilon-\delta$ definition of a limit

I finally understand the geometric intuition behind the $\epsilon-\delta$ definition of a limit, which is actually quite neat: But I'm having trouble actually using the definition to come to a ...
0
votes
3answers
111 views

Proof that there are infinitely many prime numbers

I answered a question to prove that there are infinitely many prime numbers, but I'm not sure if my attempt is right. Can somebody help me to check if my attempt is right? I would like, if I am wrong, ...
0
votes
0answers
53 views

Trying to prove that a continuous function $\,f: \mathbb{D}^2 \rightarrow \mathbb{R}^2$ has a zero using the fundamental group?

Prove: For a continuous $\;f: \mathbb{D}^2 \rightarrow \mathbb{R}^2,\;$ $\,f\left(\omega,0\right) \notin f\left(S^1\right),\,$ given a non constant function $\left(f_{\mid S^1}\right)_{\ast} : ...
1
vote
1answer
51 views

Whether a real number is a dyadic rational iff its binary expansion terminates?

In self-studying a textbook on computability theory, I found that many of the exercises depend on the following factlet: A dyadic rational is a rational number whose denominator is a power of two, ...
2
votes
2answers
92 views

Simple proof that $|xy| = |x||y|$

Apologies if this is a duplicate, I had no luck trying to find this (simple) question anywhere. Define $|x| = \max\{x,-x\}$. Prove that $|xy| = |x||y|$. This result seems incredibly intuitive, ...
1
vote
0answers
35 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
2
votes
1answer
45 views

Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.

This is Velleman's exercise 3.4.26 (b): Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$. I do understand that a number will be ...
1
vote
1answer
38 views

Proof strategy/writing for change of variables

Claim: If $f(x)=g(x)$ for all $x$, then $f(x+c)=g(x+c)$ for all x. Proof (attempt): Set $u=x-c$, and substitute $x=u+c$. $f(x)=g(x)$ implies $f(u+c)=g(u+c)$ for all $u$. Because $u$ is a dummy ...
0
votes
1answer
35 views

Proof for a first-order differential equation claim

Claim: If $p(x)$ is a solution to a first-order differential equation in the form of $df/dx=g(f)$, then $p(x+c)$, with $c$ constant, is a solution as well. I know the idea of the proof, but I am ...
0
votes
1answer
19 views

Prove that there are no integer solutions x,y to the following system of equations using mod 4 arithmetic:

So i was given a question stated in the title and I have to show this for A)$2x+7y=3$ B)$3x+ 8y = 3$ C)$4x + 9y = 5$ I understand how to use the linear diophantine equation to solve these ...
0
votes
1answer
19 views

Proof that the fractional knapsack problem exhibits the greedy-choice property

I have the following problem: Prove that the fractional knapsack problem has the greedy-choice property. The greedy choice property should be the following: An optimal solution to a problem ...
1
vote
3answers
56 views

How do I know when to use “let” and “suppose” in a proof?

When the goal is $∀n\in\Bbb N ∀m\in\Bbb N (n \ge m \rightarrow H_n-H_m \ge {n-m \over n})$, I can begin the proof with "Let n and m be arbitrary. Suppose n and m are natural numbers" or "Let n and m ...
2
votes
2answers
78 views

Prove the root is less than $2^n$

A polynomial $f(x)$ of degree $n$ such that coefficient of $x^k$ is $a_k$. Another constructed polynomial $g(x)$ of degree $n$ is present such that the coefficeint of $x^k$ is $\frac{a_k}{2^k-1}$. ...
0
votes
3answers
35 views

For any two sets A and B, if $f: A \rightarrow B$ is injective, then if A is countable, B must be countable.

So i was given two questions you either prove or disprove them. A) For any two sets A and B, if $f: A \rightarrow B$ is injective, then if A is countable, B must be countable. B) For any two sets A ...
1
vote
1answer
22 views

Determining Injectivity, surjectivity, bijectivity, and inverses

I was given a question that begins like this. Suppose that $A$ is the set $\{a,b,c\}$ (these are just names for some three elements - you don't know anything about $a,b,$ or $c$). Consider the ...
3
votes
4answers
84 views

Fibonacci proof question $\displaystyle \sum_{i=1}^nF_i = F_{n+2} - 1$

The sequence of numbers $F_n$ for $n \in N$ defined below are called the Fibonnaci numbers. $F_1 = F_2 = 1$, and for $n \geq 2$, $F_{n+1} = F_n + F_{n-1}$. Prove the following facts about the ...
1
vote
1answer
76 views

$\mathbb{A}^2\setminus (0,0)$ is not affine

I want to prove that $X = \mathbb{A}^2\setminus (0,0)$ is not affine. My attempt: If $\Bbbk[X] = \Bbbk[x,y]$ then $X$ is not affine since $(x,y) \subset \Bbbk[x,y]$ is a proper ideal, but $V(x,y) ...
1
vote
2answers
37 views

Is this function a bijection?

$f: \Bbb N \to P(\Bbb N)$ be given by $f(n) = \{n+1,n+2,n+3,\ldots\}.$ From general intuition and reasoning I think the function is not injective here is my work. If $n = 1$ $f(1) = ...
1
vote
2answers
55 views

Let $f : \mathbb{N} → \mathcal{P}(\mathbb{N})$ be given by $f(n) = \{n+1 , n+2 , n+3 , . . . \}$

So i was given a question like this Let $ f : \Bbb N\to \mathcal P(\Bbb N) $ be given by $f(n) = \{n+1 , n+2 , n+3 , . . . \}$ (a) Is f an injection? Explain (b) Is f a surjection? Explain. I ...
0
votes
1answer
43 views

Is this alternative proof of Theorem 3.7 (“Baby” Rudin, Ch. 3) correct and, if so, well written?

Rudin, in his Principles of Mathematical Analysis, proves the following theorem: The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. I've tried to ...
3
votes
4answers
2k views

Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$

I want to prove by induction that $a^n \mid b^n$ implies that $a \mid b$ holds for all integers $n\geq 1$. Clearly for $n=1$ this is true, since if $a \mid b$, then $a \mid b$. Suppose this is true ...
1
vote
4answers
623 views

If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational [closed]

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational. I can understand how it works in my head, I don't know how to prove it though.
-1
votes
2answers
54 views

Fill in the blanks with either $∈$ or $⊆$

So was given a question that begins like this Let $A=\{ \emptyset , 1 , \{2\} , \{1 , 2\} \}$ . Fill in the blanks with either $\in$ or $\subseteq$ . $\{ 1 , \{2\} \}$______ $P(A)$ ...
2
votes
5answers
50 views

Prove that $(A-B) \cap (A-C) = A \cap (B \cup C)^c$ for any three sets A, B, C.

I was given a question that says Prove that $(A-B) \cap (A-C) = A \cap (B \cup C)^c$ for any three sets A, B, C. I'm completely lost with this question. In a previous question that says $A \cap C ...
0
votes
1answer
18 views

Linear Diophantine equation in two variables

So I was given a question to find if there is any integer solutions. $6x + 15y = 79, x,y \in \Bbb Z$ Proof $3(2x + 5y) = 79$ implies 3|79 which is absurd because no such x,y exist Then I was given ...
0
votes
1answer
80 views

Show there are infinitely many primes that are equivalent to 1 mod 8.

Show there are infinitely many primes that are equivalent to $1 \pmod{8}$. Hello there! I have been trying to do this problem for a pretty long time with no avail. I noticed that this is really ...
0
votes
2answers
27 views

Determining bijectivity of a function

I was given a function from $f: \Bbb R \rightarrow \Bbb R \\f(x) = x^5 - 3\\$ I know this function is bijective because it is one to one, and onto. Then the question changes to $f: \Bbb Z \rightarrow ...
1
vote
2answers
28 views

Listing all elements of a set [duplicate]

I was given a question like the following: Let $A = \Bbb Z$, $B = [-1,\pi]$ , $C=(2,7)$. List all Elements of $A \cap (B^c \cap C)$. I do not really understand how to got about this problem. I ...
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 ...
0
votes
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 ...
4
votes
2answers
100 views

Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.

Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for. I must prove the following using mathematical induction: For ...