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.

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0
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1answer
43 views

Help with proof for solving an ODE using contraction mapping theorem

I'm trying to follow a proof for solving the ODE $$\frac{df}{dx} = (f(x)+x)x$$ For $0 \leq x \leq 1$ with the initial condition $ f(0)=0$. The proof I am following goes like this Define ...
3
votes
5answers
86 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 ...
6
votes
1answer
98 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
32 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?
9
votes
3answers
332 views

Prove by Hilbert deduction: $\vdash _{HFOL} \forall x (\neg(A \to \neg B))\to \neg(\forall xA \to \neg(\forall xB))$

I'd really like your help proving: $\vdash_{HFOL} \forall x (\neg(A \to \neg B))\to \neg(\forall xA \to \neg(\forall xB))$ Where $HFOL$ is the proof system which contains the Hilbert relevant ...
3
votes
2answers
83 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 $n$ ...
1
vote
2answers
67 views

Cantor-Schroder-Bernstein Contradiction

I need help figuring out where to start a proof that says I should use a proof by contradiction. $f\colon A\to B$ and $g\colon B\to A$ be functions and each is 1-1. Let $D$ be the range of $f$ (i.e., ...
8
votes
2answers
225 views

I feel the need to prove every result for myself

I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I ...
3
votes
2answers
59 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 ...
2
votes
2answers
31 views

Proving a theorem about the limit of a function

The theorem is as follows: $$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$ ...
3
votes
1answer
122 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

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
26 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 ...
0
votes
1answer
26 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 ...
2
votes
2answers
21 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 ...
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
74 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
2answers
21 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.
2
votes
0answers
101 views

Finding minimum graph of $k$ disjoints component

Sorry for the english, I tried to make it the most clear possible. Be $G$ a connected graph not directed, I have to find an algorithm that given $n$ the quantity of vertex and $0<k<n$ disjoint ...
0
votes
1answer
32 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} ...
1
vote
3answers
91 views

Is there a simple way to prove the Four Colour Theorem?

The four colour theorem says that: Given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the map so that no two ...
0
votes
1answer
37 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
60 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 $\epsilon$ ...
3
votes
1answer
124 views

Struggling with writing logical proofs

I am struggling with the way to write a clear and mathematical proof of logical theorems. Take for example the theorem $\Gamma \models A, \Gamma \subseteq \Delta$ implies $\Delta \models A$. I can ...
1
vote
2answers
60 views

Let $G$ be a connected graph. If $G$ has no cut vertices, then G has no bridges.

Prove, disprove, or give a counterexample: Let $G$ be a connected graph. If $G$ has no cut vertices, then $G$ has no bridges.
0
votes
1answer
29 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 ...
-3
votes
1answer
33 views

Proof of Theorem of Divergent Sequences [closed]

Let's say $(a_n)$ and $(b_n)$ are divergent sequences. Show or disprove the following: $((a_n b_n)_n)$ is divergent. $((a_n + b_n)_n)$ is divergent. $((c b_n)_n)$ with $c\neq0$ is divergent. ...
2
votes
1answer
895 views

Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$

Prove that $\int_a^b f(x)\,dx \gt 0$ if $f \geq 0$ for all $x \in [a,b]$ and $f$ is continuous at $x_0 \in [a,b]$ and $f(x_0) \gt 0$ EDIT. Please ignore below. It is very confusing actually -.- ...
4
votes
7answers
616 views

How do I show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab?

I really need help with this question. I am required to show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab, and I'm not quite sure how. Any work/help is ...
0
votes
0answers
40 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 $|y|\ge y$, ...
2
votes
1answer
155 views

Prove that $\lim_{\Delta x\to 0} \frac{\Delta ^{n}f(x)}{\Delta x^{n}} = f^{(n)}(x).$

If $\Delta f(x)=f(x+\Delta x)-f(x)$, $(a)$ prove that $$\Delta\{\Delta f(x)\}=\Delta^2f(x)=f(x+2\Delta x)-2f(x+\Delta x)+f(x);$$ $(b)$ derive an expression for $\Delta^n f(x)$ where $n$ is any ...
0
votes
1answer
33 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.
3
votes
3answers
81 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.
0
votes
1answer
25 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 ...
0
votes
1answer
33 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!
2
votes
1answer
36 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 ...
1
vote
0answers
21 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 ...
2
votes
1answer
23 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 ...
1
vote
1answer
44 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} ...
2
votes
0answers
74 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 ...
0
votes
1answer
29 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 ...
0
votes
1answer
18 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
45 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 ...
1
vote
2answers
38 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
0answers
111 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 ...
0
votes
1answer
20 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
31 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
27 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 ...
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 ...
13
votes
7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...