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
4answers
52 views

Finding growth bounds on Fibonacci Sequence

I've been working on this following problem: Find a constant $c< 1$ such that $F_n \leq 2^{cn}$ for all $n \geq 0$. I honestly have no idea where to begin on this. I've done plenty of proofs ...
3
votes
1answer
49 views

Prove that $R$ is an equivalence on $\mathscr P(A)$. Is this correct?

Suppose $B\subseteq A$, and define a relation R on $\mathscr{P}(A)$ as follows: $$R=\{(X,Y)\in\mathscr{P}(A) \times \mathscr{P}(A)\mid(X\mathrel{\triangle} Y)\subseteq B\}$$ Prove that $R$ is an ...
2
votes
4answers
87 views

Prove that if $3|mn$, then $3|m$ or $3|n$

I am trying to prove this for integers $m$ and $n$. I tried to reach prove that $3|m$ by assuming that 3 does not divide $n$, but this is such a basic assumption of mine already that it is hard for ...
1
vote
2answers
74 views

prove P ∧ Q → P ⇔ R ∨ ¬R in natural deduction

I am a beginner in Natural Deduction currently reading the book "Logic in Computer Science" and got stuck at proving: $$ P\land Q\to P\Leftrightarrow R\lor\lnot R$$ The latter formula is clearly a ...
0
votes
2answers
23 views

Proving the summation of a function as big theta of another function

Show that $\sum^n_i i^4\log^2i$ = $\Theta(n^5\log^2n)$ I am completely lost on how to solve this problem. I understand that $\Theta$ deals with the upper and lower bounds, so do we prove both big-oh ...
0
votes
0answers
28 views

Deeply confused while trying to understand the derivation of infinite product representation of gamma function

I am trying to understand how in the world Euler figured out the infinite product representation of Gamma function. $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$ Of ...
5
votes
1answer
40 views

Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
1
vote
0answers
44 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
1
vote
1answer
25 views

Relative Primes and Congruence

Suppose that $a$ and $n$ are relatively prime. Prove that there is an integer $b$ such that $ab\equiv 1\pmod n$ .
0
votes
0answers
98 views

Squeeze theorem and $\frac{\sin x}{x}$

I've been going over old calculus books to refresh my memory and have mainly been focusing on proofs. One of the things I found interesting was the squeeze theorem, even though since basic calculus i ...
1
vote
2answers
62 views

Finding an operation on $G^S$ that yields a group

Problem: Assume $S$ is a nonempty set and $G$ is a group. Let $G^S$ denote the set of all mappings from $S$ to $G$. Find an operation on $G^S$ that will yield a group. Update (full attempted ...
3
votes
1answer
26 views

Prove: R∩R−1 is symmetric.

The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric." If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, ...
2
votes
2answers
73 views

Proving a Simple Fact about Slopes of Lines

The following problem is a detail from a proof I wrote recently -- a detail that I left unproven, and would like to prove. Let there be three points $a$, $b$, and $c = \frac{a+b}{2}$, with $a<b$. ...
1
vote
2answers
52 views

Prime Factorization

Let $n\ge0$. What is the power of $2$ in the prime factorization of $(2^n)!\,$? Prove that the value is correct. So far I've conjectured the value to be $2^n - 1$. This is true for $n=0,1,2,3,4$. ...
0
votes
1answer
31 views

generalised eigenspace is invariant proof

let $T \in \text{End}(V)$ where $V$ is a finite dimension vector space. We define $V_j(\lambda) = \ker ((\lambda I - T)^j))$ as the generalised eigenspace. I am trying to prove that this space of is ...
0
votes
1answer
30 views

Let $I,J,K$ ideals of ring R. Prove that $I+JK \subseteq (I+J)(I+K)$

Let $I,J,K$ ideals of ring R. Prove that $I+JK \subseteq (I+J)(I+K)$ Comments: This is something I need to solve an exercise. Previously tasted the ring I'm working $IJ = I \cap J$ should the need ...
2
votes
3answers
46 views

Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is 1 or 2 if $m$ and $n$ are relatively prime.

Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is $1$ or $2$ if $m$ and $n$ are relatively prime natural numbers. Can anyone give a step-by-step answer for this?
1
vote
1answer
46 views

Prove that a ring does not have a multiplicative identity

Let $R =$ the set of all matrices $ \left[ \begin{array}{cc} x & 0 \\ y & 0 \\ \end{array} \right] $ where $x, y \in \mathbb Z$ with R being a ring under matrix addition and ...
1
vote
2answers
43 views

Let G be a simple graph of order $n\geq 2$. If $|E(G)|>\binom{n-1}{2}$,then G is connected.

Let G be a simple graph of order $n\geq 2$. If $|E(G)|>\binom{n-1}{2}$,then G is connected. One of the solution I get is as shown as below: Suppose G is not connected, Then G is a disjoint union ...
3
votes
10answers
395 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
0
votes
3answers
33 views

How can I prove the inverse of a function is odd?

Given function $f$, where $A⊆ \mathbb{R} $ is a symmetric domain with respect to 0,$ \;\; f:A \rightarrow\mathbb{R}$ and $f$ is an odd one-to-one function, I need to prove $f^{-1}$ is odd. I was ...
1
vote
2answers
51 views

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$ "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss ...
1
vote
0answers
26 views

I need help with a proof in Probability [duplicate]

I could use help with the following problem: Show that if $A$, $B$ and $C$ are three events such that $P(A \cap B \cap C) \neq 0$ and $P(C \mid A \cap B)$ = $P(A \mid B)$, then $P(C) = P(A \mid B)$. ...
3
votes
1answer
44 views

How to prove $(\forall x,y\in\mathbb{Z})(5\nmid xy\to(5\nmid x\land 5\nmid y))$

Question: Prove $x,y\in\mathbb{Z},\Bigl((5\nmid xy)\to(5\nmid x\land 5\nmid y)\Bigr)$ where $\forall a,b\in\mathbb{Z},\bigl((a\nmid b)\leftrightarrow(\forall k\in\mathbb{Z},b\neq ak)\bigr)$ and ...
0
votes
0answers
38 views

Existence of non-coprime between an integer and an arithmetic sequence

Take two relatively prime numbers $m,n \in \mathbb{Z}$ (i.e. $gcd(m,n) = 1$) where $m \neq 1$. Show that: $$\forall a \in \mathbb{Z} \textrm{ s.t. } 0<a<n$$ $$\exists i \in \mathbb{Z^+} ...
-1
votes
5answers
86 views

Prove that the sequence $\sin\left(\frac{n\pi}{3}\right)$ diverges

I don't want to hear that since $sin$ is a periodic function, etc, then we are done. I would like to see a simple proof that make use of the definition of convergence of a sequence. I have tried to ...
0
votes
1answer
30 views

Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative ...
4
votes
1answer
106 views

Existence of bijection that reorders elements?

Suppose I have some function $f:\mathbb{R}\to[0,1]$. Does there necessarily exist a bijective mapping $g:\mathbb{R}\to\mathbb{R}$ such that $g(x)\leq g(y)$ implies $f(x)≤f(y)$? If not, does it help if ...
1
vote
2answers
31 views

how to prove the uniqueness and existence of equations

I've the equation $e^x=5$, know it has the solution $x=\ln 5$. How to prove the existence before, and after the uniqueness of this solution?
0
votes
2answers
29 views

Discrete Math Combinatorics, permutation, one-to-one proof

I am having trouble getting started with the following proof: (This is homework, so I'd appreciate a nudge in the right direction.) Let m, r $\in$ N with 0 $\leq$ r $\leq$ m. Prove that the number of ...
1
vote
3answers
17 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
0
votes
2answers
53 views

If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$

Just wanted to confirm that this is a correct solution: Proof: Suppose $f(x_0) > 0$ for some $x_0 \in [a,b]$. Then, by continuity of $f$, for $\epsilon < f(x_0)$, there exists $\delta > 0$ ...
0
votes
0answers
47 views

Conjugacy Class Equation

$\newcommand{\cl}{\operatorname{cl}}$Let $p$ be a prime and let $G$ be a group with $|G|=p^n$. Show that $Z(G)\neq\{e\}$. The class equation states $|G| = |Z(G)| + |\cl(a_1)|+\cdots+|\cl(a_n)|$ where ...
3
votes
7answers
217 views

Which number is bigger: $\sqrt[10]{2}$ or $1.2$?

What is the general method for finding such inequalities? I have some more problems of this kind in the text I am using.
0
votes
1answer
26 views

Sequential Criterion for Continuity

I want to prove the following: Let $A$ be a nonempty subset of $\mathbb{R}$, $c\in A$, and $f : A \to \mathbb{R}$. Then $f$ is continuous at $c$ if and only if for every sequence $(x_n)$ in $A$ such ...
6
votes
2answers
77 views

Prove that $ (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally connected and connected subspace of $\mathbb{R}^2$

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that $A = (\mathbb{Q}\times\mathbb{R})\cup (\mathbb{R}\times\mathbb{Q})$ is a locally ...
0
votes
0answers
7 views

What does $\rho_A\left(\frac{1}{t}\right)$ mean, if the function is the characteristic polynomial?

Given $A=\mathbb{R}^{n\times n}$ and its characteristic polynomial $\rho_A(t)$ show that $q(A^{-1})=0$ where the function is defined as: $$q(t)=\frac{1}{\rho_A(0)}t^n\rho_A\left(\frac{1}{t}\right)$$ ...
0
votes
0answers
28 views

Deciphering whether a set relying on a predicate exists

I'm having some trouble with these style of questions in my Set Theory course and I'm not sure how to proceed with them. Screenshot of the question For part i) I tried something along the lines of ...
2
votes
2answers
45 views

Prove that if $\gcd(m,n)=1$ then every divisor $d|mn$ has a unique form $d=ab$ such that $a|n$ and $b|m$.

I can see why this is true. I have a problem with formality or with explaining certain things properly. An attempt: suppose there are two forms $d=a_na_m=b_nb_m$ such that $a_n,b_n|n,a_m,b_m|m$ but ...
1
vote
0answers
39 views

Product over real interval? Is there a better way of putting this?

In my amateur interest, I have arrived at this (nothing rigorous here at all):$$\prod_{a\in [1,2]} \prod_{b=0}^\infty f(a,b) \neq 0$$ For starters, there might be a more intuitive way about doing ...
1
vote
3answers
83 views

Prove that a subset is normal

Let $H$ be the subset of $GL(2, \mathbb R)$ consisting of all matrices of the form $$ \left[ \begin{array}{cc} x & 0\\ 0 & x\\ \end{array} \right] $$ where $x \neq 0$ Prove that H ...
4
votes
5answers
278 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
0
votes
1answer
22 views

$\sigma$-algebra with cardinality $\aleph_0$ [duplicate]

Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals? I do not have a clue on how to start with this? Can someone please give me a hint?
0
votes
3answers
66 views

Beginner linear algebra proof

Hello I am a bit confused on trying to prove a result in linear algebra. I will most what I know but I think it is very incomplete etc. $\mathbf{Thereom}:$ Let $A \in \mathbb{M_{nxn}}$, and v an ...
2
votes
3answers
36 views

If $f$ is a uniformly continuous function show that $g = f(x) - f(y)$ is uniformly continuous on all of $\mathbb{R}^2$

Problem statement: Let $f: \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function on $\mathbb{R}$ and let $g: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x) - f(y)$. Then $g$ is uniformly ...
1
vote
0answers
32 views

help me to prove this

I would like to prove that $n/k!$ for $k$ that holds $k\leq \frac{\log n}{\log\log n}$ will always be bigger/equal to $1/2$. I tried to use stirling but got stuck. Any ideas? Thanks, Jonatan
2
votes
2answers
53 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
1
vote
3answers
24 views

Set Theory Proof with Complements

If $A \cap B = \emptyset$ then $A \subset B'$ and $B \subset A'$, where the prime symbol denotes the complement of each set. Here are my thoughts: Assume $A \cap B = \emptyset,$ since the ...
1
vote
2answers
41 views

Faithful Group Actions and Normal Subgroups

Let $G\curvearrowright X$. Show that $K=\{g\in G:g\cdot x=x,\text{ for all }x\in X \}\trianglelefteq G$. If $\phi\colon G\to Sym(X)$ is the homomorphism given by the action, show that $K=\ker(\phi)$. ...
0
votes
0answers
43 views

What is the sum of reciprocals of Natural Numbers? [duplicate]

I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...