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
0answers
26 views

Simple Turing machine problems [duplicate]

I'm trying to go over some review problems regarding Turing Machine recognizability, and am still pretty confused about the following problems. This is the only information we are given in the problem ...
1
vote
1answer
43 views

Distributing identical objects into distinct boxes

The problem I'm trying to solve is: find the number of ways of distributing $r$ identical objects into $n$ distinct boxes such that no box is empty, where $r \geq n$. I've found conflicting answers ...
1
vote
1answer
37 views

Splitting parties into committees

I feel like this should be an extremely simple problem, but I can't quite figure it out. How many ways are there to split $2n + 1$ places in a committee among $3$ nonempty parties, such that a ...
3
votes
3answers
55 views

Baby Rudin Exercise 4.2

Can someone check my proof? If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $$f(\overline{E}) \subset \overline{f(E)} $$ for every set $E\subset X$. ...
3
votes
1answer
99 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
1
vote
1answer
43 views

Explain this proof by induction? [duplicate]

$P(n)$ is the statement $n! < n^n$, where $n$ is an integer greater than $1$. I found a solution online here (https://people.cs.umass.edu/~barring/cs2... But I don't understand how they got from ...
0
votes
1answer
83 views

Density of Subgraphs

I am stuck trying to make sense of this review problem: Given a graph G(V, E), we say that the induced subgraph G(S) on a subset of vertices S ⊆ V is a subgraph of G whose vertex set is S and edge ...
0
votes
3answers
153 views

Proving prime number combinatorics

I am trying to figure out the following review problem: Let $p$ be a prime number and $a$ be a natural number. Prove that the following (parts 1, 2, 3 and 4) are true for every $p$ and $a$. Here, ...
0
votes
1answer
105 views

Turing Machine recognizability

I'm trying to go over some review problems regarding Turing Machine recognizability, and am still pretty confused about the following problems. This is the only information we are given in the problem ...
1
vote
1answer
104 views

Proving that Unit Intersection is NP-complete

I am extremely stuck on how to go about this problem and any help would be so appreciated. We are told to consider the following combinatorial problem: Unit Intersection: Let X = {1, 2,...,n}. ...
0
votes
1answer
22 views

Use the Pumping Lemma to prove that $L=\{a^{m}b^{2^{n}} \}$ is not regular.

Normally pumping lemma proofs aren't too hard; you just experiment with different values of $i$ and sort of stumble on the solution. However, this problem is really bugging me and I am spending a lot ...
0
votes
1answer
12 views

Linear Second order Differential operator proof questions

I have 3 proof questions from my book that I have tried and I would like to see if my solutions are valid and/or there is a simpler way to prove them. Firstly, the notation $ker(L)$ means all $f$ ...
0
votes
1answer
23 views

Application of Implicit Function Thm

Problem Let $f_{1},f_{2}$: $R^{2}\rightarrow R$ of class $C^2$. Consider the zero sets $Z_{1}, Z_{2}$ (of $f_{1},f_{2}$ respectively) ie $Z_{i}=\{(x,y) | f_{i}(x,y)=0\}$. Assume $\nabla f_{i}(x,y) ...
1
vote
1answer
32 views

Quick proof help dealing with the intersection of sets

For each real number $r \in (1, 3)$, define $A_r$ to be the interval $[0, r)$. Set $B = \bigcap A_r$. Prove that $B = [0, 1]$. I understand this problem up until the part where we have to set $r$ ...
2
votes
2answers
48 views

Finding conditions to make roots of a quadratic less than one in magnitude

I'm doing a problem that asks for you to find the conditions that make $y$ defined: $$y=x^2-bx+c$$ have real roots with magnitude less than one. Now the condition for the roots being real seems to ...
3
votes
1answer
31 views

Proving the equivalence of a finite set

Let A be a finite set. Prove that if A≈􏰔n and A≈􏰔m, then n=m. The answer in the book uses a max function, so I was just wondering if there was a simpler way. If not, it would be appreciated if ...
3
votes
3answers
24 views

Let $A$ be a finite simply ordered set.

Show that $A$ has a largest element. [Hint: Proceed by induction on cardinality of $A$] Attempt: According to the assumption my set $A$ is finite and simply ordered so that would mean $A = \{A_1, ...
1
vote
3answers
37 views

Is there anything to prove in this corollary?

Show that if $B$ is not finite and $B\subset A$, then A is not finite. I mean the statement is very trivial, but I'm having an issue actually writing what I would deem a good proof of this. The only ...
40
votes
9answers
7k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
0
votes
1answer
35 views

Complex Analysis: Exhibiting an upper bound

Let $f:D(0,1) \to \mathbb{C}$ be an analytic function such that $|f(z)| \leq M, ~\forall z \in D(0,1)$ and $f(z_1) = 0.$ Claim: The estimate \begin{equation*} |f(z)| \leq M \left( ...
0
votes
1answer
39 views

Group Theory-Isomorphisms

Currently in Abstract Algebra, discussing group theory. In order to show two groups are isomorphic to each other, I know what you need to show, $1$-$1$, onto, and homomorphism. what I'm having a ...
0
votes
0answers
54 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
1
vote
1answer
71 views

How do I use the principle of mathematical induction to prove whether or not $\sum_{k=1}^n (-1)^k = \frac{(-1)^n-1}2$ is a true statement?

For all n elements of Natural Numbers,$\sum_{k=1}^n (-1)^k= \frac{(-1)^n-1}2$. I proved p(1) to be true : $\sum_{k=1}^1 (-1)^k = (-1)^1 =-1$. And $\frac{(-1)^1-1}2 = \frac{(-2)}2 = -1$ So P(1) ...
6
votes
1answer
77 views

Prove this Complicated Inequality

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$ ...
1
vote
3answers
52 views

Let $x_n$ be a sequence. Prove that $x_n \rightarrow 0 $ iff $(x_n)^{2} \rightarrow 0 $

Let $x_n$ be a sequence. Prove that $x_n \rightarrow 0 $ iff $ (x_n)^{2} \rightarrow 0 $ Attempt Assume that $(x_n)^{2}$ converges to zero. So $| x_n|| x_n| \lt \epsilon'$ after some stage. Thus $| ...
2
votes
4answers
76 views

Proving that $A\subset B$ if given $A=A\cap B$

Let $A = A \cap B$. Prove $A \subseteq B$ I go about like this : Let $x \in (A \cup B)$ $\implies x\in A ~~\text{and} ~~ x\in B$ Question 1 : Is this true? Will and come here? Ideally or ...
0
votes
1answer
26 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
0
votes
1answer
30 views

Proof of Partial Sum - Series

By considering the partial sums for $S$, that is $$S_n =1+2+3+···+n$$ show that the infinite series $S$ does not converge. How do we show that this does not converge? How to rigorously prove it ...
1
vote
0answers
27 views

Series Proof Question [duplicate]

By considering the partial sums for S, that is Sn =1+2+3+···n show that the infinite series S does not converge. However in this video http://www.numberphile.com/videos/analytical_continuation1.html ...
0
votes
1answer
37 views

Complex analysis exercise - boundary points of nonconstant analytic functions.

The exercise has two parts: a) Suppose $f$ is nonconstant and analytic on $S$ and $f(S)=T$. Show that if $f(z)$ is a boundary point of $T$, $z$ is a boundary point of $S$. b) Let $f(z)=z^2$ on the set ...
3
votes
8answers
111 views

Proving that $12^n + 2(5^{n-1})$ is a multiple of 7 for $n\geq 1$ by induction

Prove by induction that $12^n + 2(5^{n-1})$ is a multiple of $7$. Here's where I am right now: Assume $n= k $ is correct: $$12^k+2(5^{k-1}) = 7k.$$ Let $n= k+1 $: $$12^{k+1} + 2(5^k)$$ ...
0
votes
1answer
24 views

Let T be a one-to-one linear transformation from $R^m$ to $R^n$ and B={$e_1$,$e_2$,…,$e_m$} a basis for $R^m$.

Prove that the set {T($e_1$),T($e_2$),...,T($e_m$)} is an independent set. Let T : $R^n$ → $R^m$ be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in ...
1
vote
1answer
40 views

Proving that a punctured disk is not simply connected, using a specific definition

I am dealing with the same set based on my previous question. I want to show that the set $H = \{z \in \mathbb{C} : 0 < |z| < 1\}$ is NOT simply connected, using the following definitions, that ...
3
votes
3answers
37 views

Prove that Euclid's algorithm computes the GCD of any pair of nonnegative integers

I've been struggling with a basic exercise involving Euclid's algorithm and mathematical induction. Given the following definition of the Euclid's algorithm (in Java): ...
0
votes
0answers
8 views

Combinatorial proofs with vandermond's identity [duplicate]

I am studying for my final for discrete math and I have come across a proof that I am confused on solving. I was wondering if anyone could help. I understand that it is vandermond's identity but I ...
2
votes
1answer
49 views

Proof for existence of LU decomposition

The LU Decomposition for a matrix exists if and only if it can be reduced to reduced row echelon form without any row interchanges. How to prove this theorem?
0
votes
1answer
36 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
1
vote
1answer
37 views

Injections from a set of functions to R

Show there is an injection from $\Bbb R^2 \to \Bbb R $ Does there exist an injection from $X \to \Bbb R$ where $X $ is the set of all functions where f(x)=x for all but finitely many x. This is a ...
4
votes
3answers
428 views

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$.

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$. I am very new to proofs and not completely sure of how to approach this one. I tried several different ...
3
votes
0answers
31 views

Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
0
votes
1answer
48 views

Show that there exists a unique function with a certain property

I'm trying to prove the following theorem: "Let $~f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}~$ be a function, and let $~c~$ be a natural number. Show that there exists a unique function $~a: ...
1
vote
3answers
67 views

No continuous injective map $f: \mathbb{S}^1 \to \mathbb{R}$ [duplicate]

A friend asked me if there could be a continuous injective map $$f: \mathbb{S}^1 \to \mathbb{R}.$$ My intuition tells me no. Endow $\mathbb{S}^1$ with a topology $\mathscr{T}$ and fix a pole $x ...
0
votes
2answers
57 views

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to $\sqrt x$ [duplicate]

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to c. I know I have to estimate $| \sqrt x_n - \sqrt x |$. But I cannot start. Thanks
0
votes
3answers
27 views

Cornered sphere homemorphic to unit sphere

A buddy of mine asked me this question, to which I found it intuitively obvious, but was unable to come up with a proper proof. Consider the so-called cornered sphere, defined by $x^4+y^4+z^4=1$ in ...
4
votes
3answers
39 views

Let $a \leq x_{n} \leq b$ for all n in N. If $x_{n} \rightarrow x$. Then prove that $a \leq x \leq b$

Let $a \leq x_{n} \leq b$ for all n in N. If $x_{n} \rightarrow x$. Then prove that $a \leq x \leq b$ Attempt - If I assume that $x$ is greater than both $a$ and $b$. Then since series is given ...
0
votes
1answer
25 views

Proving $S_1 \subseteq S_2$ for transitive closure

This is one of the problem I have been working from Velleman's How to prove book: Suppose $R_1$ and $R_2$ are relations on $A$ and $R_1 \subset R_2$l (a) Let $S_1$ and $S_2$ be the reflexive ...
0
votes
2answers
29 views

Using induction prove $\sum\limits_{k=1}^n\dbinom{k}{k-1}$=$\binom{n+1}{n-1}$

$\sum\limits_{k=1}^n\dbinom{k}{k-1}$=$\binom{n+1}{n-1}$ We are supposed to use induction to prove this inequality. After the base case, I tried to use the definition $\binom{n}{k} = ...
0
votes
1answer
16 views

More clarification on an equivalence relation problem already answered

So this problem already has a solution: Problem with Equivalence Relations I'm good for the majority of it except for part c), I wasn't able to figure that out on my own or by looking at the ...
2
votes
2answers
72 views

Proofs of theorems, where picture is sufficient

A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is ...
0
votes
0answers
31 views

multivaraible chain rule proof

I wanted to prove the multivaraible chain rule; I had to prove that $df \large \frac {({x(t)},{y(t)})}{dt} = \frac{∂ f}{∂ x}\cdot\frac{dx}{dt} + \frac{∂ f}{∂ y}\cdot\frac{dy}{dt}$ So, I took the ...