# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $W \in \mathcal{P}(A)$ ...
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### 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 ...
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### Relative Primes and Congruence

Suppose that $a$ and $n$ are relatively prime. Prove that there is an integer $b$ such that $ab\equiv 1\pmod n$ .
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $a<b$. ...
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### 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 $n=0,1,2,3,4$. ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $P(C) = P(A \mid B)$. ...
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### 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 ...
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### 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 ...
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### $\sigma$-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?
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### 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 ...
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### 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 ...
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### 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
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### 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. ...
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### 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 ...
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 $K=\ker(\phi)$. ...
I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...