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
3answers
23 views

Help with partitions, equivalence classes, equivalence relations.

The following definitions and results are from my textbook. A partition $\mathcal{P}$ of a set $X$ is a collection of nonempty sets $X_1, X_2, \dots$ such that $X_1 \cap X_j = \emptyset$ for $i ...
3
votes
1answer
45 views

What does the notation $11\mid a^2$ mean?

What does the notation $11\mid a^2$ mean as used in this answer: http://math.stackexchange.com/a/948251/13230 I am trying to understand the proof that $\sqrt{11}$ is an irrational number, but am ...
0
votes
1answer
23 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
0
votes
0answers
14 views

Proof of optimal substructure for the “plus sign game”

First of all, I think there's no "plus sign game", I have just invented the name to describe the problem faster. Another thing: I thought to ask the question in these stack exchange's website because ...
1
vote
1answer
19 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
0
votes
1answer
49 views

Proof of a Vector Space

Let $F$ be a field and let $(V, +, F)$ be a vector space over $F$. If $W_1$ and $W_2$ are subspaces of $F$, prove that $W_1 - W_2 = \{v \in V | v = w_1 - w_2 \text{ for some } w_1 \in W_1, w_2 \in W_2 ...
1
vote
1answer
38 views

Exercise from (Baby) Rudin (Chapter 3, exercise 13): is this proof correct? Is it well-written?

The problem is the following: Prove that the Cauchy product of two absolutely convergent series converges absolutely. Here is my attempt: Let $s_n=\sum^n_{k=0}a_k$ and $t_n=\sum^n_{k=0}b_k$ be ...
1
vote
1answer
13 views

Maximum maintains order under limit in $\mathbb{R}^2_{+}$

I'm trying to show that if: $$ (a_{1n},a_{2n})\to (a_1,a_2)\\ (b_{1n},b_{2n})\to (b_1,b_2)\\ max\{a_{1n},a_{2n}\}\geq max\{b_{1n},b_{2n}\},\forall n\in\mathbb{N} $$ Then: $$ max\{a_1,a_2\}\geq ...
0
votes
2answers
21 views

Clarification: Prove there exists a number $N$ such that $n > N$ implies $s_n >a$

Below is the proof that I have been working on and the solution provided by the professor. Let $(s_n)$ be a convergent sequence, and suppose $\lim s_n > a$. Prove there exists a number $N$ ...
0
votes
1answer
43 views

About proof writing in axiomatic set theory

I meet question as following: i) Show that the mappings $f: X \rightarrow Y$ from one given set $X$ into another given set $Y$ themselves form a set $M(X, Y)$. ii) Verify that if $R$ is a set ...
0
votes
0answers
6 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
55 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
26 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$$X \cap (Y \setminus Z) = (X \cap Y) \setminus (X \cap Z)?$$
1
vote
1answer
73 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 ...
3
votes
1answer
22 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
48 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
68 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
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 ...
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
68 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
113 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
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 ...
1
vote
1answer
52 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
36 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
90 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
22 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
57 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
36 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
23 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
57 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
46 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
56 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
53 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 ...