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.

learn more… | top users | synonyms

0
votes
1answer
290 views

provide a combinatorial proof that $C_{n+1} = C_0C_n + C_1C_{n-1} + …. + C_kC_{n-k} + …C_nC_0$

(a) Let $C_n$ denote the number of ways of writing a valid list of open and closed parentheses of length $2n$ (valid means that at any point along the list, the number of open parentheses must be ...
0
votes
0answers
73 views

The poor relation? Time to reassess?

The idea of functions is one of the most important ideas in mathematics. It rules mathematics: one input gives one output. Although, in game theory one study multi-valued functions: one state of the ...
1
vote
1answer
32 views

Four-point geometry proof

I'm new to writing proofs and am working with proving finite geometry systems. I'm not sure how I should answer this one. Using the four point finite geometry system: prove that there exists a set of ...
1
vote
1answer
20 views

Find the positive integer $n$ such that $p(z)=z^5-30z^2+1$ has exactly $3$ zeros(counting multiplicity) in $\{z \in \mathbb{C}:n<|z|<n+1\} $

Could anyone advise me on how to find the positive integer $n$ such that $p(z)=z^5-30z^2+1$ has exactly $3$ zeros(counting multiplicity) in $\{z \in \mathbb{C}:n<|z|<n+1\} \ ?$ Hints will ...
1
vote
1answer
55 views

K[$\alpha$]/K is algebraic

I have to prove that: If $\alpha \in E$ and $E/K$ is a field extension and $\alpha$ is algebraic over $K$ then $K[\alpha]/K$ is algebraic, but I do not know how to use the hypothesis to do this. Can ...
0
votes
1answer
67 views

Prove this recurrence relation? (catalan numbers)

$$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots + C_kC_{n−k} + \cdots + C_nC_0\text{ ?}$$ Where Cn denotes the number of ways of writing a valid list of open and closed parentheses of length ...
3
votes
2answers
43 views

A problem on fixed point

Let $f$ be holomorphic function defined on a domain which contains the closed unit disk $\overline {D(0,1)}.$ Suppose $f$ maps $\overline {D(0,1)}$ into open unit disk $D(0,1).$ Could anyone advise me ...
1
vote
2answers
49 views

How to learn developing the formal proof of statements or theorems?

This is not a math problem, but something that this platform can provide answers to. I am a student of math. I learn and understand the concepts. However, when I see some statements I understand how ...
1
vote
0answers
19 views

Injectivity of Theta functions

Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian Theta functions (notations like on wikipedia). $F:=\left\{ \tau \in \mathbb{C}: Im(\tau)>0, \left| Re(\tau)\right|<1, ...
0
votes
1answer
31 views

If $P$ is a polynomial of degree $n>0$, then there exists circle $C$ of radius $R$ such that $\int_{C} \frac{P^{\prime}(z)}{P(z)} dz=2n\pi i $

Let $P$ be a polynomial of degree $n>0.$ Could anyone advise me on how to show there exists $R>0$ such that if $C$ is the circle $|z|=R$ anticlockwise oriented, then $\begin{align}\int_{C} ...
1
vote
1answer
22 views

Why is the equality right? (Set-Theory)

Let $A, B$ finite sets, and let $f,g\in A\to B$. Also, Let the equivalence class: $$f \sim g \iff \exists h\in Eq(A,A). f=g\circ h $$ Claim: $$f\sim g \iff \forall b\in B. \left| \left\{ a\in A : ...
1
vote
2answers
24 views

What is the cardinality of $\left|C_s\right|$?

Let $$C_s = \left\{ f\in \mathbb{N}/S \to \mathbb{N} : \forall M\in \mathbb{N} / S. f(M)\in M \right\}$$ Where $S$ is an equivalence class. I need to prove $$\left|C_s\right| > \aleph_0 \implies ...
1
vote
1answer
41 views

Segner's Recurrence Relation [closed]

Why is Segner's Recurrence Relation formula valid. Does anyone know how to prove it? I can't seem to understand why this formula works/is true. $$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots ...
0
votes
2answers
57 views

Suppose that S is a nonempty set such that A= LUB S. Let T={2x+1:x element in S}. Prove that LUB T=2A+1

Can some one please help me get started on this. I would really like to understand what I am doing. Suppose that S is a nonempty set such that A= LUB S. Let $T={2x+1:x \in S}$. Prove that LUB ...
1
vote
1answer
97 views

Proof of recursive formula for Catalan numbers, and their interpretation as the number of paths

If $C_n$ is the $n$th Catalan number, then show that they satisfy the following recurrence: $$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots + C_kC_{n−k} + \cdots + C_nC_0\text{ ?}$$ I tried ...
3
votes
0answers
57 views
+50

Seeking help extending Vieta-jumping to higher powers

I am trying to prove the following conjecture. Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} ...
0
votes
1answer
45 views

How do I start this proof?

How can I begin proving this? I am not sure where to start. A(S) is defined to be the set of all bijections from S to S.
0
votes
1answer
57 views

How to show that $\lim_{n \to \infty} \frac{\binom{n+k}{k}}{(n+k)^k} = \frac{1}{k!}$?

Given that $k\in\mathbb{N}$, my question is how to prove that this sequence converge to $\frac{1}{k!}$: $$\left\{ \frac{n+k \choose k}{(n+k)^{k}} \right\}_{n\in\mathbb{N}}.$$ I have this attempt:
2
votes
2answers
70 views

Cauchy sequences.

Well my question is how to prove this: Let $a_{0}$, $a_{1}$ be distinct real numbers.Define: $$a_{n}=\frac{a_{n-1}+a_{n-2}}{2}$$ for each positive integer $n\ge2$.Show that $\{a_{n}\}$ is a Cauchy ...
0
votes
2answers
53 views

I need help with a limit proof

I tried to proof this limit but when i get epsilon i can't narrow it because there's a factorization something unusual. The limit that i need to proof is: $\lim_{z\to1}\frac{{z^2 -1}}{z-1} = 2$ I ...
0
votes
1answer
31 views

How many strings are there (inclusion exclusion principle)

Q: What is the number of strings with the length of $8$ above $\left\{1,2,\cdots,10\right\}$ where $7,8$ appears at least one time? So by using the inclusion exclusion principle: $10^8$ ...
1
vote
1answer
22 views

If a graph $G=\left<V,E\right>$ is connected and $\left|E\right|=\left|V\right|$ then there's a circle in $G$

Prove: If a graph, $G =\left<V,E\right>$ is connected and $|E|=|V|$ then there's a cycle in $G$ I think this should be proved by induction. This is certainly holds for $n=1$ (a vertex with ...
3
votes
1answer
20 views

Going from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨~q)

I am confused on how to go from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨ ~q). I know they are equal because I plugged them into a truth table and all of the rows have the same values. What would be some ...
2
votes
1answer
216 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
1
vote
1answer
29 views

Lattice theory question

I am having trouble with the following question Show that a lattice is distributive iff for any element $a,b,c$ in the lattice $$(a\lor b)\land c \leq a \lor(b\lor c)$$ My attempt: Let the ...
0
votes
1answer
17 views

What assumption is needed to add together equations in a vector space?

Suppose in a vector space $V$, we have $x+z=y+z$. We want to show that $x=y$. Note, by one of the vector space axioms, for all $z \in V$ there exists an additive inverse $v \in V$ such that for all ...
0
votes
3answers
44 views

Meaning of specific part of answer in delta-epsilon limit proof

In a math question I had such as the limit approaches $3$ in $x^4-x^2+1$ (help with formatting please I'm new), the answer was that $\delta$ equals $\left(1, \frac{\epsilon}{168}\right)$. In another ...
0
votes
1answer
64 views

Pigeonhole proof of the existence of two numbers with given sum [duplicate]

Let $|W|=m+1$ and $W$ be a subset of $X=\{1,2,3,\dots ,2m\}$ ($m$ is any natural number). Prove there exists two numbers in $W$ whose sum is $2m+1$. Can anyone give me a hint to prove this? I ...
2
votes
2answers
22 views

Basic Properties Explanation

In regards to divisibility I am having trouble wrapping my head around some of the concepts, more specifically some of the general properties of divisibility. for example, why is it possible for ...
1
vote
1answer
49 views

Application of Opening Mapping theorem

Let $f$ be a holomorphic function on open set $A$ such that $(Im(f(z))^3 + (Re(f(z))^4 =5.$ Could anyone advise me on how to use Open mapping theorem to prove $f$ is constant? Hints will suffice. ...
1
vote
1answer
21 views

Proving a graph is connected by minimum degree

Let number of vertices be >= 2. If the min degree of G >= n/2, then the graph is connected. I was trying to solve this using contradiction, but now I'm stuck. So I started out with "If the the min ...
0
votes
1answer
53 views

geometry circle proof

Use a common notion to prove the following result: If P and Q are any points on a circle with center O and radius OA then OP is congruent to OQ. Since O is the center and P & Q are any where on ...
0
votes
1answer
26 views

Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle ...
1
vote
1answer
61 views

Proof by cases. Formulate a conjecture. I don't get it. Question inside.

I don't understand this math question for my discrete math 2 class. FOrmulate a conjecture about the decimal digits that appear as the final decimal digit of the fourth power of an integer. Prove ...
4
votes
1answer
46 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
-1
votes
6answers
40 views

Proving $2^{2n}-1$ is divisible by $3$ for $n\ge 1$

So I decided to use induction. First, I started with my base case, $P(1) = 2^{2(1)}-1=3,$ so it's true. That means if $n = k$ is true, then $n = k+1$ is true also. So, $P(n+1)-P(n)$ would also be ...
15
votes
10answers
2k views

Having hard time understanding proofs by contradiction.

I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. $\textbf{Theorem:}$ If $P \rightarrow ...
6
votes
3answers
100 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
0
votes
0answers
59 views

Similar Triangles Proof - How to tackle proofs?

Well, I know it is repetitive.I have read the proof from different textbooks.But sometimes I feel doubtful about it all.Every time I try to prove it for myself, I fail at some points.I'm asking those ...
1
vote
3answers
35 views

A number to the group cardinality power

Well my question is how is possible this: Consider an element $g\in G$, where $G$ is a finite group, then you have: $g^{|G|}=e$ How can I prove it? Thank you.
3
votes
2answers
219 views

Proof of trigonometric identity using vector calculus

Question: Using vector calculus, show that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ I have no idea how to even attempt the question. A small hint to help me get started would be greatly ...
1
vote
2answers
106 views

An example of set with a countably infinite set of accumulation points

I have to give An example of set with a countably infinite set of accumulation points, and I say: We can consider the set or real numbers and we take an arbitrary real number $x$ then the interval ...
0
votes
0answers
28 views

LUB, GLB, maximum and minimum of a set .

I am not sure if my solution for the following problem is correct. Evaluate LUB, GLB, maximum and minimum (if they exist) of $\{-n: n ∈ \Bbb N\}$. My answers: LUB: $-1$ GLB: $-\infty$ max: $-1$ ...
2
votes
4answers
61 views

Exotic proofs of $\sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$

Let $p,n$ be positive integers. The following identity $\displaystyle \sum_{j=0}^{n-1}\binom{p+j}{p}=\binom{p+n}{p+1}$ may be proved by induction or by successive uses of Pascal's rule (both ...
0
votes
3answers
44 views

Beginner Question about the definition of finite sets

Hi this question is in regard to a part of chapter 7 in Daniel Velleman's book "How to Prove it". I just began to learn about infinite sets and such and there is one part that confuses me. It starts ...
2
votes
1answer
47 views

Showing two functions are uniformly continuous

I have no idea how to prove this detail (uniformly continuous) about these functions because they're defined to $\infty$. I need the general mindset to prove it, or any ideas. Thanks in advance. $$ ...
0
votes
1answer
32 views

Why does this proof make sense and theorems required

I saw the proofs on the derivative of $\frac{d e^x}{dx}=e^x$ from here and the one that was intriguing was this : $$e:=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n \implies \frac{d(e^x)}{dx} = ...
0
votes
0answers
31 views

Properties of Sup, Inf on sets.

I understand this proof is a bit long-winded, but I am only concerned with it is correct or not. It seems sound to me. Claim: If $A,B \subset \mathbb{R}$ and are non-empty, $a \leq b, \forall a \in ...
1
vote
0answers
45 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
0
votes
1answer
72 views

Show that an entire function that is real only on the real axis has at most one zero, without the argument principle

Could someone advise me on how to approach this problem: Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. without the use of argument principle ...