For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

learn more… | top users | synonyms

6
votes
1answer
166 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
0
votes
1answer
38 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
3
votes
5answers
102 views

Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
3
votes
2answers
120 views

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
3
votes
2answers
67 views

Is this a proof by contradiction?

Below is a proof that any group of order $p^2$ is abelian $(p$ prime of course). Let $Z \left({G}\right)$ be the center of $G$. We know $|Z(G)|>1$. $\color{blue}{\text{Suppose}} \left\vert{Z ...
3
votes
1answer
129 views

Why is this more-detailed proof more acceptable than its trivial counterpart?

Say that we're asked to give a proof of 'proof by induction'. i.e. for some property $P$, proving that $$\forall n,P(1) \wedge [P(k) \implies P(k+1)] \implies \forall n, P(n)$$. Now, I understand ...
2
votes
2answers
37 views

Proving a theorem about the limit of a function

The theorem is as follows: $$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$ ...
2
votes
2answers
45 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
0
votes
1answer
29 views

Proof homomorphism between graphs

Given two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, an homomorphism of $G_1$ to $G_2$ is a function $f:V_1 \rightarrow V_2$ such $(v,w) \in E_1 \rightarrow (f(v),f(w)) \in E_2$. We establish that ...
0
votes
1answer
32 views

Proof that a local minimum in a spanning tree is also a minimum spanning tree.

Be $G$ a connected graph with weights associated to its edges. Be $T(G)$ the graph that has the spanning trees of $G$ as vertex, and two spanning trees are adjacent to each other if and only if each ...
2
votes
2answers
25 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
0
votes
2answers
23 views

Help proving this Proposition

For every natural number $n$, the integer $6^{2n+1}+8^{3n}$ is divisible by 7. I handled the base case quite well, but got stuck on the induction step. Any help would be greatly appreciated.
1
vote
2answers
57 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
3
votes
1answer
84 views

How to prove l'Hospital's rule for $\infty/\infty$

I'm having trouble with this l'Hospital's rule wiki page(the proof of l'Hospital's rule): http://en.wikipedia.org/wiki/LHospital%27s_rule Well, in the case where the limit looks like $0/0$, it's ...
0
votes
1answer
34 views

If $|G|=p^n$ for prime $p$, then $|\mathcal{Z} (G)|\neq p^{n-1}$

I am trying to prove the following: Let $|G|=p^n$ for $n\geq 1$ and $p$ prime. Prove that $|\mathcal{Z} (G)|\neq p^{n-1}$. Here is what I have so far: Suppose, to the contrary, that $|\mathcal{Z} ...
0
votes
1answer
41 views

Question about the Least squares method

We have $n$ dots: $(x_1,y_1)\cdots (x_n,y_n)$. We know that if we use the Least squares method we will get a line $y=mx+b$ that giving the minimal value for the function $w=\sum_{i=1}^n ...
0
votes
1answer
113 views

Epsilon delta proof

I know that the point of the proof is to show that you can get within $\epsilon$ of the limit, by giving a value that is within $\delta$ of $x$. But when solving for $\delta$ in terms of $\epsilon$ ...
0
votes
0answers
61 views

Mathematical logic and proofs involving absolute values

Is the following proof correct? Let ($\forall$ x, y $\in $$\Bbb R)$ $|x-y| \le |x| +|y|$ case#1: Suppose x and y $\ge0$. We want to show that $|x-y| \le |x| + |y|$. Since $|x|\ge x$ and $|y|\ge y$, ...
0
votes
1answer
37 views

Are these proofs correct?

I haven't formally learn how to do proofs, but I attempted some of these. It'd be great if you guys can check them and give me some pointers. Thanks!
3
votes
3answers
91 views

If a connected graph has a unique spanning tree, then it is a tree.

Prove if a connected graph has a unique spanning tree, then it is a tree. Edit: This can be shown with proof by contradiction.
2
votes
1answer
39 views

Is this proof by induction correct?

Prove by induction that for all $n\in\mathbb N$, $3\mid n^3+3n^2+2n$. $$P(1) = (1)^3+3(1)^2+2(1) = 6$$ Which is clearly divisble by $3$. Therefore, $P(1)$ is true. Assume $P(1),\ldots,P(n)$ and ...
0
votes
1answer
42 views

Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$

Prove, disprove, or give a counterexample: Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$. Eccentricity - the ...
0
votes
1answer
34 views

If a tree has order 2 or more, then the minimum cut set is 1. [closed]

Prove: If a tree has order 2 or more, then the minimum cut set is 1.
0
votes
1answer
26 views

Suppose $f:X\to Y$ is onto and $A\subseteq Y$. Then $f(f^{-1}(A))=A$.

Prove, disprove, or give a counterexample: Suppose $f:X\to Y$ is onto and $A\subseteq Y$. Then $f(f^{-1}(A))=A$. Edit: Does this work? Suppose $f:X \to Y$ is onto and $A \subseteq Y$. We know ...
1
vote
0answers
26 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...
1
vote
1answer
46 views

Is this equation for $2^k!$ correct?

I couldn't find any equation for $2^k!$ so I came up with an equation that appears to work for the factorial of a power of $2$. However, I'm having problems proving it. My equation: $$ \def\x{\times} ...
0
votes
1answer
64 views

Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$

I've seen looking on previous questions that using this algorithm, I can construct self-complementary graphs. I got confused at this point, because I'm not sure if I should proof that there are no ...
2
votes
1answer
26 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : *$\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$ by definition we have ...
0
votes
1answer
20 views

Help proving a theorem in my textbook

If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational. For reference, an integer $n$ is a perfect square if $n=m^2$ for some $m \in \mathbb{Z}$. Any help proving this ...
2
votes
2answers
46 views

Help with a proof my professor gave my class

Let $x,y \in \mathbb{R}$ with $x<y$. There exists an irrational number $z$ such that $x<z<y$. My proof so far: Let $x,y \in \mathbb{R}$ and assume $x<y$. Then, by Theorem 11.8 (in ...
0
votes
1answer
22 views

Equivalence relations proof

I need to prove that if $R_1$ and $R_2$ are equivalence relations on the set $A$, then $R_1\cap R_2$ is an equivalence relation. Problem is I dont know how. Please help!
0
votes
1answer
44 views

Combinatorics Proof

I am having trouble with a combinatorics proof. I need to prove that if $r$ <= $n$ then the number of $r$ - subsets of {1,...,n} is $n!$/$(n-r)!$*$r!$ I really struggle with writing proofs and ...
0
votes
1answer
75 views

Discrete Math Proofs, Partial Orders and Equivalence Relations

I am horribly stuck on $3$ proofs for my discrete math class. Any help would be greatly appreciated. Prove that if $R$ is a partial order, then $R^{-1}$ is a partial order Prove that if $R_1$ and ...
1
vote
2answers
42 views

Finding quantity of vertex of odd degree in a graph

Let $G=(V,X)$. If $G$ has n vertex, such exactly $n-1$ have odd degree, how many vertex of odd degree have $\overline {G}$. ($\overline {G}$ the complement of $G$.) So the first thing I notice is ...
2
votes
1answer
16 views

Proof that in a graph of $2$ or more vertrex, there's at least $2$ of them that have the same degree

That's what I what to prove let $G$ be an undirected graph such it has $2$ or more vertex, there's at least $2$ vertex that have the same degree. I'm almost certain that, supposing that it's possible ...
2
votes
0answers
117 views

Can you share your experience in hand waving and other informal communication regarding mathematical proofs?

I intend to write a paper that will address among other issues the informal communication between mathematicians. My point of origin is the view that every proof can be represented by a sequence of ...
3
votes
2answers
52 views

Proving limit of a sum

I need to prove the following: $$\lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} = 1$$ This is my attempt: \begin{align} \lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} & = \lim_{s ...
0
votes
1answer
32 views

Error replacing integral of f with its midpoint rule approximation

here is a question I've been banging my head against. If f is continuous on [a,b] and differentiable on (a,b), and if there is a positive real number M such that |f'(t)| is less than or equal to M ...
1
vote
2answers
46 views

What is the proper way to prove this?

First of all, here is the question I am trying to answer for context. I can see that the statement $\forall x \in \mathbb{Z} , \exists y \in \mathbb{Z}((x\leq y ) \wedge (x+y=0)) $ negates to ...
0
votes
2answers
61 views

Help with 2 questions my professor gave us

I was wondering how to solve these two proofs my professor put on the blackboard today. He said they were pretty easy but i'm still unsure how to prove them. ANy help would be greatly appreciated! ...
-1
votes
2answers
33 views

Show with an exemple that the inclusion could be real

Im trying to Solve this problem. I have started to Solve the first part, but need to show it with an example. Thankfull for help.
0
votes
1answer
20 views

Unsure how to solve this proof

I came across this in my textbook and was wondering how it could be proved. My only thought is that contradiction should be used. Thank you for any help! Suppose $L = \lim_{k \to \infty} X_k$. ...
3
votes
2answers
42 views

Disjoint Cycles and Supports

I am working though an Introduction to the theory of Groups. I have come the following exercise: "Let $\alpha = \begin{pmatrix} i_1 & i_2 & \cdots & i_r \end{pmatrix}$ and $\beta = ...
1
vote
1answer
43 views

Permutation cycles

My tasks are the following : Task 1 : Prove that $ \begin{pmatrix} 1 & 2 & \cdots & r-1 & r \end{pmatrix} = \begin{pmatrix} 2 & 3& \cdots & r & 1 \end{pmatrix} ...
0
votes
1answer
63 views

Proofing a Reachable Node Algorithm for Graphs

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
2
votes
2answers
87 views

How to complete this epsilon delta proof

Prove $\lim_{x\to 1} {2+4x \over 3} = 2$ using the epsilon delta definition of a limit. if $0 < \left|x-1\right| < \delta$ then $\left|{2+4x \over 3}-2\right| < \epsilon$ scratch work for ...
2
votes
0answers
93 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
1
vote
1answer
40 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
-1
votes
3answers
74 views

Show that $C(n,k) = C(n-1,k) + C(n-1,k-1)$ [duplicate]

I'm studying for my final for Statistics, and I want to understand literally every problem in my textbook (at least in the first 7 chapters). One of the problems asks to show that ${n}\choose {k}$ ...
2
votes
1answer
170 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...