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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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 ...
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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
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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
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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 ...
3
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2answers
49 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 ...
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1answer
28 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 ...
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2answers
43 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 ...
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3answers
58 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! ...
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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.
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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
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2answers
38 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 = ...
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1answer
38 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} ...
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1answer
54 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
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2answers
57 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
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0answers
73 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
32 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 ...
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3answers
50 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 ${n}\choose {k}$ ...
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1answer
66 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 ...
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2answers
38 views

Usage of Maximum Modulus Theorem?

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$. I was thinking that this is case where I can use the ...
1
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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.
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1answer
27 views

If the edge $xy$ is a bridge of $G$, then $x$ and $y$ are in separate components of $G$-$xy$.

If the edge $xy$ is a bridge of $G$, then $x$ and $y$ are in separate components of $G$-$xy$. I can't think of a counterexample so I am operating under the impression that it can be proved. How does ...
1
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1answer
59 views

Proof using addition and multiplication axioms

I'm working on addition and multiplication axioms of integers for discrete math. I'm trying to prove (k - m) + (m - n) = k - n. The first step I took was this ...
2
votes
2answers
52 views

Logic Proof with Natural Deduction: if I assume the antecedent, do I still have to prove the consequent?

I have the unpleasent feeling that my "proof" is dead wrong. The core of my concerns is: when I have something like A -> (B -> C) and I assumed ...
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3answers
61 views

Prove that a graph with $n$ vertices and less than $n$-1 edges, is disconnected.

Prove that if $G$ is a graph with $n$ vertices and fewer than $n$-1 edges, then $G$ is disconnected. The book I am working through uses a similar definition of "$n$ vertices and at least $n$-1 edges, ...
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2answers
45 views

Prove the following Statement is a tautology

I need to prove the following statement is a tautology [¬Q∧(P→Q)]→¬P So far this is what i have but now i am stuck any advice on further finishing this problem would be helpful. ...
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4answers
64 views

Integral Problem $\sin^6 x$. [closed]

What is the integral of $\sin^6 x$? Can some one publish it with the method it would be really helpful. Specially I want the trig identities.
0
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1answer
22 views

Proving a circuit of a graph will have an edge in common with a cycle of the same graph.

Prove, if possible, if an edge lies on a circuit in $G$, then the edge also lies on a cycle in $G$. I attempted to find a counterexample but it seems pretty evident that this statement is true. The ...
0
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1answer
25 views

Need help with compositions of relations

Prove that given relations $R_1 \subseteq A \times B$, $R_2 \subseteq B \times C$, $R_3 \subseteq C \times D$ then $(R1 \circ R2) \circ R3 = R1 \circ (R2 \circ R3)$ I don't know where exactly to ...
1
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1answer
18 views

inductive proof of binary existence

Can someone help me witha well exlpained inductive proof of option 2 problem 2. I ha e read many different ones but they dont really make sense to me and I need to know how to do this for my final. ...
1
vote
1answer
31 views

First derivative test and uniqueness of local extrema

This is the context in which my question lies. See below for the actual question. Let $f(x)$ be differentiable everywhere and have a minimum at $x^*$. Then for every $x$ in a proper neighbourhood ...
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3answers
33 views

Help solving a proof

My professor put this up on the blackboard and I was wondering how to solve it. Let $x,y \in \mathbb{R}$. Then |$x$|< |$y$| if and only if $x^2 < y^2$.
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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 ...
1
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1answer
27 views

Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $ C = [a,k,l,m]$ Show that $\backslash$ is not associative

Question : Let $A =[a,b,c,f,g,i], B=[b,f,h]$ and $ C = [a,k,l,m]$ Show that $\backslash$ is not associative by comparing $(A \backslash B) \backslash C$ with the set $A \backslash(B \backslash C)$. ...
0
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1answer
21 views

Mathematical proofs with Cardinality

Prove that for any natural number $n$, $n<$ the cardinality of continuum. Prove that Cardinality of the power sets of the naturals < the cardinality of the power set if the reals. Prove that ...
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0answers
38 views

Proof that a Polytope has vertices

As part of my Discrete Optimization course, I have a homework where I have to prove that a Polytope has vertices. I seems to have all tools in hand (definition of a vertex, polytop, convex hull, etc.) ...
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1answer
36 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
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2answers
15 views

Proving $\gcd(a,p_1p_2)>1$ $\Rightarrow$ $p_1\mid a$ xor $p_2\mid a$ using Euclid's Lemma

Consider the lemma: If $\gcd(a, p_1p_2)>1$, then either $p_1\mid a$ or $p_2\mid a$. How can this be proved using Euclid's Lemma?
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3answers
34 views

Help with this function proof

If a function is bijective then its inverse is unique. I came across this in my textbook and was wondering how it is proved. Thank you.
1
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1answer
271 views

induction proof for kleene star

i am going through some past exam paper questions on regular languages for some revision, and i am having a bit of trouble with converting general ideas into formal mathematical proofs. the question ...
1
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1answer
45 views

Question about Logic Proof

Assuming $P$ is a unary predicate and $Q$ is a propositional variable, I'm trying to prove the following implication: $$ (\forall x (P(x)\rightarrow Q)\rightarrow ((\forall x P(x) )\rightarrow Q) $$ ...
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6answers
76 views

Prove if $ab>0$ and $bc<0$ then $ax^2 + bx +c = 0$ has two real solutions

For there to be two real solutions in a quadratic equation the discriminant, $b^2-4ac$, has to be positive, so $b^2-4ac > 0$. Rearranging the equation, I get $b^2 > 4ac$. Then $b > ...
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3answers
41 views

Let $ n,m \in Z$. Prove the following: If $m$ and $n$ are even, then so is $mn.$

Question: Let $ n,m \in Z$. Prove the following: If both $m$ and $n$ are even, then so is $mn.$ Attempt: By Definition 2.3.1, we let $ n \in Z$. Then, $n$ is even whenever there exists some $ k \in ...
2
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3answers
169 views

Proving units in a ring

Suppose $R$ is a ring with no zero divisors and with identity $1_R$ not equal to $0_R$. Suppose that $a,b$ are in $R$ and that $ab$ is a unit. Prove that $b$ is a unit. My thoughts: I know a ...
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1answer
18 views

How to prove that if $X\sim P(\lambda) \Rightarrow Var(X)=\lambda$?

How to prove that if $X\sim P(\lambda) \Rightarrow Var(X)=\lambda$? $P(X)$ means: Poisson distribution. Thank you!
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0answers
26 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
3
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1answer
62 views

Spivak Ch 11 Theorem 7

Can someone please explain how to supply a rigorous $\epsilon,\delta$ argument for this theorem as Spivak says ? My argument is: $f'(a)=\lim_{h\to 0} f'(\alpha_h)$ equivalent to ...
0
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1answer
44 views

Prove by Induction that: $1+nx\le (1+x)^n$?

$1+nx\le (1+x)^n$, for all real numbers $x>-1$ and integers $n\ge 2$. Can you please also explain a little of the basic step and the inductions step. Thank you in advance.
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2answers
55 views

Proving the inscribed angle theorem

I need to prove that a circle's inscribed angle is 1/2 of the arc it intercepts. I am given that one of the chords making up the angle is the diameter. I have an entire project to do based off of this ...
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1answer
39 views

Proposed proof for Sobolev space result

I have the following result which seems that it must be true, but I would like to prove it: This is my proposed proof. If $U \subset \mathbb{R}^{n}$. Given $u \in W^{1,p}(U)$, where $u$ has compact ...
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2answers
42 views

one to one positive integers and positive rationals

How would you go about proving that there is a 1 : 1 correspondence between the set of positive integers and the set of positive rationals. I know there are a lot of ways to do this but I am looking ...