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
2answers
14 views

Proving a Relation that is a Function by Division Algorithm

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
32 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
21 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
6 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
28 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
64 views

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 power ...
3
votes
1answer
29 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
38 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
28 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
30 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? and we ...
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
67 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
38 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
26 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
vote
2answers
54 views

Show that Y=aX+b is an random variable.

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
34 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
65 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
21 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
10 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
41 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
32 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
34 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
19 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 ...
0
votes
0answers
13 views

Construct a bijection $f:(U(Z/(8),\times) \to G$ such that $f([3]) =g_1, f([5]) =g_2$.

Let G be a group of order $|G| = 4$ such that $g^2 = e_G$ for every $ g\in G$.Take two elements $g_i \in G\ {e_G}(i = 1,2).$ (i) Construct a bijection $$f:(U(Z/(8),\times) \to G$$ such that $$f([3]) ...
0
votes
0answers
18 views

Bijective functions on a finite set

Suppose that A is a finite set and f : A → A and g : A → A are functions. I need to prove that g ◦ f is a bijection if and only if f and g are bijections. So, could I say: Assume g of f is a ...
0
votes
1answer
23 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
1
vote
1answer
20 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
votes
2answers
51 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
0
votes
2answers
27 views

continuity of functions on intervals

Suppose that $f : (a,b) \to \mathbb R$ is continuous. Then, there is a continuous $g : [a,b] \to \mathbb R$ such that $g(x) = f(x)$ for all $x \in (a,b)$. That is, a function defined and continuous on ...
-2
votes
1answer
34 views

Limits and ranges of functions

There is no function $f : \mathbb R \to \mathbb R$ that is continuous on $\mathbb R$ and with range equals to $[-2,5) \cup (-7,-4]$. True or false? If true, prove. If false, give counterexample. I ...
0
votes
0answers
27 views

Proving a theorem about matrix derivations

Ok, so Im doing some research and I have to understand the following theorem. The theorem states: Let $h$ be a derivation on $Tn(R)$ with $h(e_{ij})=0,\,\, 1\le i \le j \le n$. Then $h=\bar\delta$ ...
0
votes
0answers
14 views

Classification of surface with 18-gon planar diagram

For starters, this is a problem from L. Christine Kinsey's "Topology of Surfaces." The problem is to classify the surface using cut and paste arguments on polygons. However, between my limited ...
0
votes
3answers
30 views

Functions of sequences and convergence

(a) If $f$ is continuous on $[0,\infty)$ and {$x_n$} is a sequence in $(0,\infty)$ such that {$f(x_n)$} diverges to $\infty$, then $\lim_{n \to \infty} x_n = \infty$. (b) If $f$ is continuous on ...
0
votes
2answers
24 views

continuity and sequences

If $f$ is continuous on $[a,b]$ and {${x_n}$} is a sequence in $(a,b)$, then {$f$(${x_n}$)} has a convergent subsequence. True or False? If true, prove. If false, give a counterexample. I'm guessing ...
0
votes
0answers
17 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
0
votes
1answer
42 views

Continuous functions on closed and open intervals

(a) If f is continuous on $[a,b]$ and $f(x) > 0$ for all $x \in [a,b]$, then there is some constant $C > 0$ such that $f(x) \geq C$ for all $x \in [a,b]$. (b) If f is continuous on $(a,b)$ and ...
3
votes
2answers
38 views

GCD Direct Proof

I need to show that if $a,b,c$ are ints such that $\gcd(a,b) = 1$ and $c|(a+b)$, then $\gcd(c,a) = \gcd(c,b) = 1$ I want to try and prove this directly because I think it will be more straightforward ...
1
vote
2answers
42 views

I.V.T Continuity proof

If $f$ is defined on $[a,b]$ and has the property that, for any $k$ between $f(a)$ and $f(b)$, there is some $c \in (a,b)$ such that $f(c) = k$, then $f$ must be continuous on $[a,b]$. True or False? ...
1
vote
0answers
40 views

Working with the Mobius transformatios and linear algebra.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to $M$ is the map: $z \to ...
-3
votes
2answers
39 views

How to show $a c\equiv bc\pmod{n}$ where $n \ge 2$ does not imply $a \equiv b \pmod n$? [on hold]

How to show $a c\equiv bc\pmod{n}$ where $n \ge 2$ does not imply $a \equiv b \pmod n$? Would it be possible for someone to go over this step by step?
0
votes
0answers
20 views

Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...