For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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-2
votes
3answers
33 views

Prove/ Disprove; if a is divisible by bc, then a is not divisible by b and a is not divisible by c [closed]

The way I am currently trying is using the contrapositive, so $a\mid b$ or $a\mid c$ $\implies$ $a\nmid bc$ so I am not sure how to prove this
0
votes
0answers
30 views

Proofs needed for certain results related to functional equations

Today our maths teacher told us the following results without stating the proofs: (These are all polynomial or exponential functions) 1) $f(x+y)=f(x)+f(y)$ then $f(x)=kx$ 2) $f(x+y)=f(x)f(y)$ then ...
0
votes
1answer
27 views

Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [closed]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
0
votes
1answer
29 views

Understanding density of irrational numbers and Archemedian property

From Density of irrationals I know this much of the proof of the density of irrational numbers "We know that $y-x>0$. By the Archimedean property, there exists a positive integer $n$ such ...
1
vote
1answer
26 views

How to prove this inference in sequent calculus?

I'm using the event-B prover to proove some proof obligations. I have a relation representing a $table: table \in 1‥n \to \mathbb{N}$. I know that in a sorted table the following property is true: ...
2
votes
2answers
48 views

Proving $\log n < \sqrt n$

I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof: ...
2
votes
1answer
23 views

Side-angle-side and side-angle-angle as proved by Euclid in the Elements (Proposition 26)

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html To prove that one side is equal to another, Euclid assumes that one side is bigger ...
4
votes
2answers
69 views

Help with a proof of pairs of real numbers

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a > b$ and $c > d$ with $a \neq c$ or $b \neq d$, such that $$\left|\frac{a-b}{c-d} - 1\right| < \frac ...
0
votes
2answers
25 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 $a$ ...
0
votes
2answers
49 views

If A is countable $\mathbb{R} \setminus A$ is dense. Clarify one line in proof? Ways to improve?

Here's the proof I was given: Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ and $b$, with $a < b$, such ...
1
vote
0answers
21 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
0answers
25 views

Explain why $I$ is a function from $P$ to $P$ and determine whether it is one-to-one and onto.

The question and the solution are:( uploaded a photo so it is easier to see the formulas) So I am confused about the formula of p(x). P is the set of polynomial of x. OK, but why it makes p(x) = ...
2
votes
0answers
33 views

How should I learn the Mathematical Proofs?

S.E advisers, What is the most efficient way to learn the basic proof methodologies, which are essential for studying the mathematical analysis and number theory? I am very interested in studying ...
-1
votes
1answer
29 views

Prove that: If $n$ is not divisible by $5$, then $n^2$ is not divisible by $5$ [closed]

Suppose that $n$ is not divisible by $5$, then $n^2$ is not divisible by $5$. I tried using contrapositive to prove this, but I don't know how to proceed.
1
vote
2answers
35 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 ...
1
vote
3answers
43 views

Prove that $\Gamma\left(-a\right)=\left[\Gamma\left(a\right)\right]^{-1}$ for $\Gamma:\mathbb{Z}\rightarrow \mathcal{B}\left(A,A\right)$

I am working through various problems in Bloch's Proofs and Fundamentals and I'm stuck on this problem (in need of hints): Let $A$ be a set. A $\mathbb{Z}$-action on $A$ is a function ...
1
vote
1answer
26 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
0
votes
1answer
30 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 ...
3
votes
2answers
83 views

How to prove the the addition of tangent is the same as the multiplication? [duplicate]

If A,B,C are angles of a triangle show that: $$\tan A+ \tan B+\tan C = \tan A \tan B \tan C $$ I've tried this many times but I cannot seem to prove it, can someone show me how to solve this ...
2
votes
1answer
25 views

Proving a collection of subsets is a basis

I am given this definition of a basis: Let $a$ be a point in a metric space $X$. A collection, $\mathfrak{B}_a$, of neighborhoods of $a$ is called a basis for the neighborhood system at $a$ if every ...
1
vote
4answers
32 views

Epsilon-Delta Continuity proof (verification/help)

So, I am really bad at these problems, and I don't know why. Edit: The metric over $\Bbb R$ is assumed to be $|f(a,b)-f(x_1,x_2)|$ Problem statement: Define $f: \Bbb R^2 \rightarrow \Bbb R$ by ...
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
0answers
23 views

Show: If $v \in E^{\perp}$ then it can be written as $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$

(i) Assume that $B = \{w_1,\dots,w_k\}$ is an orthogonal basis for $E$. Let $v \in E^{\perp}$ such that $v\neq O_{V}$. Prove that $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$ for some nonzero $w\notin E$ and ...
1
vote
0answers
22 views

How can I show uniqueness of a (constructed) topology.

Let $X$ be a set, and $\Phi$ a set of subsets of $X$ such that: $\varnothing, X \in \Phi$ If $\{ F_i: i\in I\}\subseteq \Phi$, then $\bigcap_{i} F_i \in \Phi$ If $F,G \in \Phi$ then ...
1
vote
3answers
18 views

Proving continuity with two different metrics

Problem statement: Let $X$ be the set of all continuous functions $f:[a,b]\rightarrow \Bbb R$, and define the metric $d^*(f,g)$ on $X$ by $$d^*(f,g) = \int_{a}^{b} |f(t) - g(t)|dt$$ Now, for each ...
0
votes
0answers
20 views

Help with a proof involving integration of difference of functions

Let $f(x,a)$ and $g(x,a)$ be two continuous functions from $[0,1]\times \mathbb{R}^+ \to \mathbb{R}$. $g(x,a)$ is decreasing and convex in $x$. $f(x,a)$ is decreasing in $x$ and increasing in $a$. ...
0
votes
2answers
11 views

Proof of show transitivity between 3 variables with exponents

If $a^5$ divides $b$ and $b^5$ divides $c,$ show that $a^{20}$ divides $c.$ Please help me prove this proposition.
1
vote
3answers
43 views

Proving an identity

Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is ...
1
vote
2answers
58 views

“Easier” way to prove $|X| < |\mathcal P \left({X}\right)|$?

(EDIT: Thanks for the counterexample. The assumption $(\forall f)(sur(f) \iff |Im(f)| = |B|)$ is the major flaw in the proof, since it's only true when $B$ is finite.) I was reading about Cantor's ...
1
vote
1answer
45 views

Show that the following map is a bijection

$$g(x,y) = \frac{1}{2}(x-1)x + y$$ where $g:\mathbb{Z}^+ \times \mathbb{Z}^+ \longrightarrow \mathbb{Z}^+$ Attempt: I am only having problems with proving the injectivity part so that is all I'm ...
3
votes
2answers
323 views

Proof by contraposition not making sense

I fail to understand why contraposition works intuitively. Take this sentence for example: $\text{If I pass my exams then I am a good student.}$ $\text{I pass my exams }\implies\text{ I am a ...
9
votes
1answer
82 views

Interesting properties of the function $(a,b)\mapsto a/(a-b)$

Consider the extremely simple function $$f(a,b)=\frac a{a-b}.$$ This gives the coordinate where the line through $(0,a)$ and $(1,b)$ meets the $x$-axis. I noticed that the function $f$ has some ...
0
votes
1answer
42 views

Prove $g(x) = x^3$ is continuous at $x_0$ arbitrary

We are proving $g(x) = x^3$ is continuous at $x_0$ arbitrary. My attempt: For all $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for an arbitrary point $x_0$ and $ \lvert x - x_0 \rvert ...
1
vote
3answers
46 views

How am I to interpret this result?

Prove of give counter example: Let A and B be sets. $$ A \backslash ( A \backslash B) = B \backslash ( B \backslash A) $$ An attempt at a proof: If $ x \in A \backslash ( A \backslash B)$ then $x ...
0
votes
1answer
59 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
2
votes
1answer
56 views

Prove $f(x) =x^2$ is continuous at $x_0=2$ using the $\epsilon$-$\delta$ definition

We want to prove $f(x) =x^2$ is continuous at $x_0=2$ using the $\epsilon$-$\delta$ definition. My attempt: We want the function $f$ to satisfy the definition of continuity, meaning : For all ...
0
votes
1answer
22 views

How to prove Thomsen's theorem?

Thomsen's theorem states that given a triangle ABC, choosing a point on AB (but not A or B) and doing the internal path parallel to AC till reaching BC, and then doing the path parallel to AB till ...
3
votes
2answers
45 views

Proving UNIT INTERSECTION NP-complete [duplicate]

I am working on some review problems right now and am extremely stuck on how to solve problem - any help would be so appreciated. We are told to consider the following combinatorial problem: Unit ...
0
votes
1answer
30 views

Overestimating in limit convergence proofs

I understand that for all $\epsilon >0$ there exists an $N \in \mathbb{R}_+$ such that $n>N$ implies $|a_n−L|<\epsilon$, for all $n>N$. But I find myself overestimating. For example, ...
1
vote
1answer
23 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
5
votes
1answer
76 views

Is concatenation of digit-strings transitive?

Given two digit-strings $A$ and $B$, let $AB$ be their concatenation. So for example, if $A = ``102"$ and $B = ``101"$, $AB = ``102101"$. We then say $AB \geq BA$ since $102101 \geq 101102$. Now ...
0
votes
1answer
19 views

Integrality conditions and proof by double counting.

Theorem $\mathbf{3.4.}$ In a block design of type $2-(v,k,\lambda)$ every element lies in precisely $r$ blocks, where $$r(k-1)=\lambda(v-1)\textit{ and }bk=vr\;.$$ The letter $r$ stands for ...
0
votes
0answers
26 views

Simple Turing machine problems [duplicate]

I'm trying to go over some review problems regarding Turing Machine recognizability, and am still pretty confused about the following problems. This is the only information we are given in the problem ...
1
vote
1answer
43 views

Distributing identical objects into distinct boxes

The problem I'm trying to solve is: find the number of ways of distributing $r$ identical objects into $n$ distinct boxes such that no box is empty, where $r \geq n$. I've found conflicting answers ...
1
vote
1answer
37 views

Splitting parties into committees

I feel like this should be an extremely simple problem, but I can't quite figure it out. How many ways are there to split $2n + 1$ places in a committee among $3$ nonempty parties, such that a ...
3
votes
3answers
55 views

Baby Rudin Exercise 4.2

Can someone check my proof? If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $$f(\overline{E}) \subset \overline{f(E)} $$ for every set $E\subset X$. ...
3
votes
1answer
99 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
1
vote
1answer
43 views

Explain this proof by induction? [duplicate]

$P(n)$ is the statement $n! < n^n$, where $n$ is an integer greater than $1$. I found a solution online here (https://people.cs.umass.edu/~barring/cs2... But I don't understand how they got from ...
0
votes
1answer
83 views

Density of Subgraphs

I am stuck trying to make sense of this review problem: Given a graph G(V, E), we say that the induced subgraph G(S) on a subset of vertices S ⊆ V is a subgraph of G whose vertex set is S and edge ...
0
votes
3answers
153 views

Proving prime number combinatorics

I am trying to figure out the following review problem: Let $p$ be a prime number and $a$ be a natural number. Prove that the following (parts 1, 2, 3 and 4) are true for every $p$ and $a$. Here, ...