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
27 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
5
votes
2answers
95 views

Idea of a proof by contradiction

Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the ...
3
votes
2answers
71 views

Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
2
votes
2answers
40 views

Prove using Hilbert calculus $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$, formal proof.

Prove: $\forall x(Px\rightarrow x\equiv a)\vdash Pb\rightarrow Pa$ Using Hilbert Calculus Format of solution: Step (my understanding) Solution: (1) $\forall x(Px\rightarrow x\equiv a)\vdash ...
2
votes
4answers
38 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
0
votes
0answers
14 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
5
votes
1answer
33 views

Proof check, showing pointwise convergence

My problem is this: For $x \in [0,\frac{\pi}{2}]$, $f_n(x) = \frac{nx}{1 + n\sin(x)}$ Find the pointwise limit of $(f_n)$ for all $x \in [0, \frac{\pi}{2}]$ I am not sure if the way I constructed ...
2
votes
1answer
24 views

Prove that if $f$ is an invertible function and $g$ is an inverse, then the codomain of $g$ is equal to the domain of $f$ and vice versa

I am trying to show, without using the bijection properties, what is above. Assume $f$ is an invertible function and $g$ is an inverse of $f$. For $f \circ g $ to be well defined then the image of ...
0
votes
1answer
19 views

Placement of quantifiers in a symbolic statement

I have the statement: Let $A_n$ be an indexed set of numbers defined by $A_n = n\mathbb{Z}_{\geq m}$ for $n,m \in \mathbb{Z}$. Consider the claim C: For all $n$, if $x,y$ is in $A_n$, then ...
-2
votes
2answers
36 views

Help with Discrete Structures proof [closed]

I don't have any clue how to do this $\sum\limits_{i=1}^n(-1)^{i-1}i^2 = (-1)^{n-1}n(n+1)/2$ whenever n is a positive integer. Please explain each step.
1
vote
2answers
39 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
13
votes
7answers
199 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
1
vote
2answers
43 views

Proving the well ordering principle

THe well ordering principle has that every subset of $\mathbb{Z}^+_0$ has a least element. or if $S$ is a non-empty subset of $\mathbb{Z}^+_0$ and $S = \{a_1, a_2, a_3 ... a_n\}$, then there is a ...
1
vote
2answers
44 views

Proof variance of Geometric Distribution

I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success. The distribution function is $P(X=x) = q^x p$ for $x=0,1,2,\ldots$ and ...
0
votes
1answer
52 views

Contradiction proofs

I'm a first year Physics student and I have some trouble approaching Proofs by Contradiction in some of my Math classes. Once I get the first 2 or 3 statements I can finish the proof but a lot of the ...
2
votes
0answers
93 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
2
votes
1answer
40 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
2
votes
2answers
54 views

On provability within minimal logic

In its most naive form my question boils down to this: when is a proposition that is provable "by contradiction" also provable "directly"? IOW, is it possible to know, a priori, that a ...
9
votes
2answers
156 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
2
votes
5answers
48 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in ...
2
votes
1answer
32 views

Injection proof

Prove that if f is injective, then $f(A \cap B) = f(A)\cap f(B)$ My answer: i) $f(A \cap B) \subset f(A) \cap f( B )$ Take an $x \in A \cap B$. $x \in A \cap B \implies x \in A \land x \in B$ $x ...
1
vote
1answer
25 views

Prove $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$

I want to show that: $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$ I started my deduction as follows: $\vdash\forall x(\alpha\to\beta)\to(\forall ...
1
vote
1answer
141 views

Eccentricity in corona product

I was studying about graph operation on wiki. Corona product of graphs $G_1$ and $G_2$, is the graph which is the disjoint union of one copy of $G_1$ and $|V_1|$ copies of $G_2$ ($|V_1|$ is the number ...
3
votes
1answer
128 views

Looking for help to clearly define a function that counts the number of twin primes in a range

My goal is to define a function that counts the number of twin primes between $q$ and $q^2$ where $q$ is any prime greater than $7$. I would like to do this using: The Sieve of Eratosthenes The ...
-2
votes
2answers
65 views

Proving that $A\cup\emptyset=A$ and $A\cap\emptyset=\emptyset$ [closed]

I need to prove that $A\cup\emptyset=A$, and $A\cap\emptyset=\emptyset$. It's seem like it's obvious, yet how can I prove it mathematically?
0
votes
1answer
52 views

The Change-making problem algorithm proof (at the dynamic programming method)

I saw here the algorithm for the "Change-making problem" (at the dynamic programming method). I saw it here: http://www.columbia.edu/~cs2035/courses/csor4231.F07/dynamic.pdf I'm trying to find a ...
1
vote
3answers
103 views

Prove $\frac{1}{n} =\frac{1}{n+1}+\frac{1}{n(n+1)}$ for all integers $n\in\Bbb Z$

I'm pretty sure that we need induction, since it's the format I had to use for previous problems similar to this (it isn't specified that it HAS to be an inductive proof, either, if there is another ...
0
votes
1answer
19 views

If $G$ is a connected graph with $n$ vertices and$ n - 1$ edges then $G$ is a tree, using Induction.

I am still new to proof methods and not sure if this is the correct use of induction. Base case: $n = 1$ has $0$ edges and is a tree. Assume every connected graph with $k$ vertices and $k-1$ edges ...
27
votes
4answers
1k views

What really is mathematical rigor? How can I be more rigorous?

I'm an undergraduate mathematics student who has received some constructive feedback from two instructors at the end of my exams. Namely, that I am a bit hand-wavey and not always very rigorous. While ...
2
votes
3answers
30 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
2
votes
2answers
67 views

Is it ok to do this change of variable in integration: let $x = x - 1$

In integrals like $\int \sqrt{x-1}\,dx$, is it ok to make this change of variable in integration: "let $x = x - 1$"? It looks sketchy — like saying, let 5 = 4.
3
votes
2answers
81 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
2
votes
1answer
50 views

Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
1
vote
1answer
33 views

An equation to prove with terms of Fibonacci sequence

I would like to prove an equation but I have stuck. The equation that is to prove is the below: $f(n)^2 + (-1)^{n+1} = f(n+1)f(n-1) , n \ge 2$. I'm trying to do an inductive proof of this equation. ...
1
vote
1answer
16 views

Proving that in a complete graph $\lambda(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\lambda(G)$ must be n-1. Since $\lambda(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Can I use the definition or should I say since ...
44
votes
9answers
8k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
1
vote
1answer
21 views

Proving that in a complete graph $\kappa(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\delta(G)$ must be n-1. Since $\kappa(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Am I approaching this proof the right way?
9
votes
2answers
12k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. ...
3
votes
3answers
100 views

Prove that if $B$ is similar to $A$, then $B^T$ is similar to $A^T$ .

If two matrix ($A$ and $B$) are similar if there exists an invertible matrix $P$, such that: $$ B=P^{-1} A P $$ I'm thinking if I can prove that $A$, $B$ , $A^T$ and $B^T$ have the same characteristic ...
0
votes
2answers
30 views

Understanding Proof that $\mathbb{R} \setminus A$ is dense. Verify proof.

Here's the proof I was given but with two minor? differences Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ ...
1
vote
0answers
27 views

Developing proof writing and logical skills

What resources can a person turn to in order to develop their proof writing and logical skills? The advanced calculus course I'm taking has made me realize how weak my logic and proof writing skills ...
0
votes
2answers
60 views

$\mathbb{R}$ is uncountable with Cantors Diagonal argument (how to improve binary expansion specificity?)

I know it's spelled out more than usual, but this is an introduction to higher math class. If there's any way I can improve this, please let me know. Thank you in advance. Let ...
5
votes
3answers
54 views

If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty $ has a limit.

This exercise is from Methods of Real Analysis by Richard Goldberg. If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty$ has a limit. I think this proof relies on the ...
1
vote
2answers
37 views

Understanding line of given proof

I have to understand a set of proofs and I don't understand the reasoning behind this line "This is an injection, if $g(b_1) = g(b_2)$ then $F_{b_1}$ And $F_{b_2}$ intersect, which we shown never ...
0
votes
1answer
27 views

Proof that the sup of a set is the least upper bound.

I have to prove: Assume $s \in \mathbb{R}$ is an upper bound for a set $A \subseteq \mathbb{R}$. Then, $s = \text{sup}A$ if and only if, for every choice of $\varepsilon > 0$, there exists an ...
0
votes
3answers
2k views

Proving a Sequence Does Not Converge

I have a sequence as such: $$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$ Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact? ...
2
votes
1answer
63 views

Proof of algorithm refinement

I recently had an interview in which I was asked to produce an algorithm to that computes the pairs of integers, from a list, that add up to a integer k. I then had to increase the time efficiency of ...
-1
votes
2answers
18 views

On Closure of Product subset of $\Bbb R×\Bbb R$

Suppose that $\Bbb R×\Bbb R$ has the standard topology. If $S=\left\{(t,\sin{\frac{1}{t}})\mid t\in R\text{ and }t\gt 0\right\}$. Show that $(0,0)$ $\in \overline{S} $
0
votes
0answers
73 views

Herbrands Algorithms and greek philospher

So the problem states "outline the steps in Herbrands algorithm leading to the proof that the following statement is right. ...
0
votes
1answer
34 views

Prove that $f^{-1}\left(U_1\times\cdots\times U_\kappa\right)=\bigcap_{i=1}^\kappa \left(f_i\right)^{-1}\left(U_i\right)$

Im working through Bloch's Proofs and Fundamentals and exercise 4.3.11 is Let $B$ be a set, let $A_i,\cdots,A_\kappa$ be sets for some $k\in\mathbb{N}$ be a subset for all ...