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
40 views

Proof by induction when numbers are to powers

Prove by mathematical induction: $$ 2^n+3^n < 5^n$$
2
votes
0answers
19 views

On the supremum of the union of two bounded sets

Let $A,B$ be bounded subsets of an ordered set $S$. Then $A \cup B$ is bounded and $\sup( A \cup B) = \sup \{ \sup A, \sup B \} $. Attempt to solution: Let $x \in A \cup B$. Then $x \in A $ or $x ...
-1
votes
2answers
22 views

Big-Oh and limits proof?

Prove or disprove: $2^n$ is in $O(3^n)$. I know I have to use some calculus limit techniques but I can't seem to get anywhere. Steps and an approach would be helpful, especially confirming if this has ...
0
votes
1answer
18 views

Proof Help for $[(A\cap B = B \cap C)\wedge (A\cup B = B \cup C)] \implies A = B$

$[(A\cap B = B \cap C)\wedge (A\cup B = B \cup C)] \implies A = B$ We must show that our conditions show that $\forall x[x\in A \rightarrow x\in B \wedge x\in B \rightarrow x \in A]$ $x \in A ...
0
votes
0answers
16 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
0
votes
1answer
26 views

Big-Omega proof using L'Hopital's Rule?

Prove or disprove: $15n^2$ is in $\Omega(3 \times 2^n)$ So we'd have to prove or disprove this statement: $$ \exists c \in\mathbb{R}^+,\,\exists B\in\mathbb{N}, \forall n \in\mathbb{N}, n ≥ B ...
0
votes
0answers
19 views

Steps: How to derive Probability density function for geometric functions

I am not from Mathematics background and hence lack awareness of many basic knowledge. So, please pardon if this sounds too trivial. I would like to know the steps with which I can obtain the pdf of ...
2
votes
2answers
31 views

Product of consecutive integers

Question 5 Prove that the product of four consecutive positive integers cannot be equal to the product of two consecutive positive integers. So it must equal $n(n+1)(n+2)(n+3)$ hence it must ...
1
vote
1answer
25 views

function of three variable is even than $f(a,b,c)=f(|a|,|b|,|c|)$

I used in the proof of Hlawka's Inequality you can find the link here Hlawka's Inequality that's if i have function of three variable is even in each variable, so that : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...
0
votes
0answers
11 views

On a certain “obvious” implication concerning odd perfect numbers

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$). ($\sigma(x)$ gives the sum of the divisors of $x$, ...
0
votes
2answers
31 views

Show that $b^{2^n}+1$ is a factor of $b^{2^m}-1$.

Let $m$ and $n$ be natural numbers such that $m>n$ and $b$ be any integer Show that $b^{2^n}+1$ is a factor of $b^{2^m}-1$.
1
vote
1answer
17 views

Help with proof about merge two heaps to one heap…

We have two heaps: $H_1,H_2$ that have $n_1,n_2$ elements ($H_1$ have $n_1$ elements and $H_2$ have $n_2$ elements). We know that the smallest element at $H_1$ is bigger the root (the biggest element) ...
2
votes
0answers
23 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
3
votes
0answers
32 views

Real analysis question: Suprema and Infima

Let $S$ be an ordered set with the $L.U.B$ Property, $S \supset B \neq \varnothing$, $B$ is bounded below. Write $L = \{ l : l \; \text{is a lower bound of } \; B \} $. Then, it follows that ...
0
votes
1answer
35 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
1
vote
1answer
30 views

Question on wikipedia's proof of rolles theorem

Here is this proof. It basically says that if a function has a maximum, then this quotient, for $h>0$ is $$\frac{f(c+h)-f(c)}{h}\le0$$ That's ok to me, because if $c$ is the maximum point, then ...
2
votes
2answers
47 views

How to prove/disprove proof on limits (delta-epsilon)

Prove or disprove: $$ \forall \epsilon > 0, \exists \delta>0, \forall x, y \in \mathbb{R}^+, |x - y| > \delta ⇒ |x + y| > \epsilon $$ I've been trying this for some time now but can't ...
2
votes
1answer
18 views

Cauchy Product $n$ times

I'm looking for a short and precise proof of the following identity; $$\left(\sum_{k=0}^\infty \frac{C_k}{k!}x^k\right)^n=\sum_{k=0}^\infty\left[ ...
-2
votes
2answers
15 views

Proving a Relation that is a Function by Division Algorithm [duplicate]

Let A=B=$\mathbb{N}$ R is: (a,b)$\in$R iff for some q$\in$Integers a=5q+b WHERE 0$\leq$b<5 Given a relation, show that it's a function. To Show: 1) $\forall$a$\in$A$\exists$b$\in$B((a,b)$\in$R) ...
0
votes
1answer
36 views

Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function? [on hold]

Let $A=B=\mathbb N$. Relation $R$ is: $(a,b)\in R$ iff for some $q \in \mathbb Z$ we have $a=5q+b$ Given a relation, show that it's a function. To Show: $\forall a \in A \ \exists b \in B$ such ...
0
votes
1answer
22 views

Diagonalisation and Kronecker Product

If $A$ is a $n\times n$ matrix with complex numbers for elements, and $C$ the $2\times2$ matrix defined by $$\begin{bmatrix} -2&4\\-3&5 \end{bmatrix}.$$ How do you prove that the Kronecker ...
0
votes
0answers
7 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
0
votes
1answer
35 views

King Arthur and knights at the round table puzzle

Can you help me with this math problem: Each of the K knights from the round table needs to choose a card which is marked with a number from 1 to N, N >= K. The cards all have different number. ...
0
votes
1answer
19 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
3
votes
0answers
76 views
+50

Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless networks specifically when calculating coverage probabilities - if needed I can provide the reference- it is powerful ...
3
votes
1answer
30 views

How to negate $\forall A. \exists a,b. a \neq b \land a,b \in P(A)$?

$$ \forall A. \exists a,b. a \neq b \land a,b \in P(A) $$ My intuition tells me it is false, because given $A=\emptyset$, then $P(\emptyset) = \{\emptyset\}$, so $a=b=\emptyset$. I proceeded to ...
0
votes
1answer
41 views

If $a+b \geq x$ is known to be true does that mean $a+b\geq x-1$ contradicts it?

So I was proving something and I'm wondering if this line of argument is correct. Suppose that it is true that given conditions $M,N,O$; $a+b\geq x$. That is given those conditions the minimum value ...
1
vote
2answers
29 views

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$. Assume polynomial $p(x)\in F[x]$ s.t. $p(r^2)=0$ If $r\in K$ and $r^2$ is ...
-1
votes
1answer
39 views

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$ , where $P$ is Lebesgue measure. What ever we do there are infinite irrational numbers for every two rational numbers, right? ...
0
votes
3answers
38 views

Proof by minimum counter example

I need to prove that $n^4-n^2$ is divisible by 12 by minimum counter example. I understand the process but I don't understand how we arrive at m>=7. I have seen different proofs but I still don't know ...
5
votes
1answer
71 views

Suppose that $[G:H]$ is a prime integer, and that $g \notin H$. Prove that H is normal in G.

Let H be a subgroup of a group G. Let $k,g \in G$ such that $gH = Hk$. Suppose further that $[G:H]$ is a prime integer, and that $g \notin H$. Prove that H is normal in G. I have totally no idea at ...
0
votes
2answers
39 views

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$ I really have no idea how to prove this problem. I started to consider: Assume $\sqrt[m]{p}\in ...
2
votes
2answers
28 views

Dimension Field True/False.

I'm having trouble approaching how to determine truthfulness and falsehood of the following type of problems. $F$ and $K$ are fields. 1) Suppose that $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ ...
-1
votes
2answers
64 views

Show that Y=aX+b is an random variable. [on hold]

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
3
votes
2answers
35 views

What natural numbers are not equal to the sum of the sum and the product of two natural numbers

What natural numbers $n$ do not satisfy the equation $$n = (x+y)+xy$$ where $x$ and $y$ are both natural numbers?
7
votes
3answers
70 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
1
vote
3answers
22 views

Limit of the function: x if x is rational and -x if x is irrational

The question is given as follows: Let $$g(x) = \begin{cases} x & x\text{ rational} \\ -x & x\text{ irrational}\end{cases}.$$ Prove that $\lim_{x\to 0}g(x) = 0.$ My first thought is to use ...
0
votes
1answer
35 views

Prove a functions is injective

Prove the function $f:\mathbb{N} \to\mathbb{N}$defined by $f(x)=2^x$ for all $x$ in $\mathbb{N}$ is one to one. Is my proof correct and if not what errors are there. For all $x_1,x_2$ $\in$$N$, ...
2
votes
1answer
61 views

Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
1
vote
2answers
22 views

How can “homotopy lifting theorem” be applied to prove this theorem?

Homotopy lifting theorem Let $p:C\rightarrow X$ be a covering map. Let $F:Y\times[0,1]\rightarrow X$ be a continuous function. Let $f:Y\rightarrow C$ be a continuous function such ...
0
votes
0answers
11 views

What theorems are used in this following proof of derivatives of log normalizer is moments of sufficient statistics?

The below is the derivation of the proof that shows derivative of log normalizer of exponential family is moments of sufficient statistics \begin{equation} ...
3
votes
0answers
36 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
0
votes
2answers
17 views

GCD proof using fundamental theorem of arithmetic

prove: $\gcd(m,n)=1$ if and only if $\gcd(m^i,n^r)=1$ I believe you need to do something with fundamental theorem of arithmetic to prove one of the sides. Not quite sure though. Help is appreciated. ...
-2
votes
1answer
33 views

Combinatorial Argument Proof

Prove: $c(40,5) = c(17,5) + c(17,4) + c(23,1) +...+ c(23,5)$ where c is the binomial coefficient. Can I use a combinatorial argument to prove?
0
votes
2answers
42 views

Proof using Induction

Give the induction proof of: $$ 1.2 + 2.3 + k(k+1) = \frac{k(k+1)(k+2)}{3} $$ Is this proof even possible? Not sure how to do.
0
votes
2answers
34 views

Generators of the Borel $\sigma$-algebra on $\mathbb{R}^2$

How do I show that a set of closed sets (plus the empty set) is a generator for $\mathbb{B}_2$? The set in question is the set made of set of vectors in a given range of angles and lengths, think of ...
0
votes
1answer
26 views

Algebra with set notation and set properties

Suppose that $S$ and $T$ are sets with $S \cap T = \emptyset$ Let $C \subseteq S \cup T$ and let $A = C \cap S$ and $B = C \cap T$. Show that $A \subseteq S$, and $B \subseteq T$. I said, let ...
0
votes
2answers
19 views

Define f : Z/3Z → Z/3Z by f ([a]) = [2a + 1].

For this problem, I have to prove the function is well-defined, is surjective, and is injective. For seeing it is well defined, I have this: Assume [a1] = [a2] in the set of equivalence classes Z/3Z. ...
0
votes
1answer
35 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
-1
votes
0answers
20 views

Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b).

So, as stated above, here is my question: Suppose f: A → B. Define a relation ∼f on A by a ∼f b if f(a) = f(b). First, I have to prove that ~f is an equivalence relation on A. So I need to show that ...