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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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
1answer
40 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 ...
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 ...
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 $f: S\to \mathbb{N}$. ...
0
votes
1answer
50 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 ...
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
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 ...
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} : ...
0
votes
2answers
95 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\frac{a_k}{2^k-1}$. ...
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
54 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 ...
-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
1answer
28 views

Help with Spivak's Calculus Chapter 3 Problem 6

It says: Show that the straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$. Since the slope in a graph of a line is determined by using similar triangles, ...
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 ...
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
27 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 ...
-1
votes
1answer
43 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
39 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
70 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
24 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
509 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
98 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 ...