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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3
votes
1answer
41 views

Prove that $\langle S\rangle$ $= G$, with $S \subset G$ and #$S > 1/p $ #$G$

I'm having trouble with solving this problem: Let $G$ be a finite group of order $> 1$, and $S \subset G$ a subset of $G$, with #$S > 1/p $ #$G$ Where p is the smallest prime factor of the ...
2
votes
1answer
29 views

Another topology question

This is a two part question. The first part, part (i), I present with the solution I reached. The second part, part (ii) is where I need help. (i) Let $B$ be a basis for a topology $T$ on a non-empty ...
1
vote
1answer
55 views

Topology related question

Salam everyone. If I understand correctly it's site etiquette to typeset math questions in tex? If that is not the case please let me know. Otherwise here is the question : Let $C[0,1]$ be the set of ...
0
votes
0answers
27 views

Please help show that the linear transformation of a subspace is equal to itself.

Let $U$ be an orthogonal $n\times n$ matrix, and consider the linear transformation $T : \mathbb{R}^n \to \mathbb{R}^n$ defined by $T(x) = Ux$. Let $W$ be a subspace of $\mathbb{R}^n$ such that $T(W) ...
0
votes
3answers
42 views

Prove that if $x^2+y^2 = z^2$ then $x$ or $y$ is even

I am having trouble proving this. I feel that proof by contradiction would be the best method, although I quickly got stuck after $x=(2k+1), y=(2j+1)$. I expanded so that $4j^2+4k^2+4j+4k+2=z^2$ but I ...
1
vote
2answers
44 views

Prove that $x_{n+2} := \frac{1}{2}(x_n + x_{n+1})$ converges, if $x_0 = 1$ and $x_1 = 2$?

This question is related to my other question, where I had just to find the limit (which is $\frac{5}{3}$) of the following defined sequence: $$x_0 = 1 \\ \\ x_1 = 2 \\ \\ x_{n + 2} = \frac{1}{2} ...
0
votes
3answers
68 views

How to prove that $z_n = 2^n$ converges and therefore has a limit?

I have to prove that the following sequence converges and therefore has a limit: $$z_n = 2^n$$ for $n \in \mathbb{N}$. I have tried to prove it, but I am not seeing exactly what I am doing, that's ...
0
votes
1answer
50 views

Prove: $<S>$ $= G$, and every $x \in G$ can be written as $x = s_{1}s_{2}$ with $s_{1}, s_{2} \in S$

I'm trying to solve this problem for my math study, but the things I'm trying don't seem to work. Let $G$ be a finite group, and $S \subset G$ a subset of $G$, with #$S > 1/2 $#$G$ Prove: a) ...
2
votes
2answers
17 views

Proving or Disproving statements using sets

I just don't seem to get proofs or set theory so hopefully my question makes sense. I'm not sure when I should or shouldn't use an example to prove or disprove a statement? One example question is, ...
3
votes
1answer
79 views

Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.

Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for. I must prove the following using mathematical induction: For ...
3
votes
2answers
64 views

2014 Putnam A1 Prime number factorial help

Question: Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number. ...
4
votes
2answers
45 views

Show that the sum of the $x$-coordinates of three points on the graph of $y = x^2$ whose normal lines intersect at a common point is $0$.

Suppose that three points on the graph of $y = x^2$ have the property that their normal lines intersect at a common point. Show that the sum of their $x$-coordinates is $0$. I've done a bit of work ...
0
votes
1answer
37 views

Prove by contradiction that a circle chord is no longer than its diameter

Can anyone help me with this homework question of mine? I'm actually new to discrete mathematics and to be specific, with proofs. Here's the question, "Prove, by contradiction, that no chord of a ...
0
votes
1answer
39 views

Group Theory - Cyclic Groups & Direct Product

Prove that if $G$ is an infinite group and $H$ is a group then $G \times H$ is cyclic if and only if $G$ is cyclic a $H =$ {${e_H}$}. Solution: I can see that this is true but I don't know how to ...
0
votes
0answers
42 views

How to acquire Mathematical Reasoning & Proof Skills

Dear Math Stack Exchange advisers, I am going to start self-studying the introductory analysis soon by using the textbooks called "Understanding Analysis" by Abbott and "Mathematical Analysis" by ...
1
vote
2answers
89 views

Prove that, for $s$ is upper bound of A, $s = \sup A$ iff , if $r < s$, so there exists $x \in A$ such that $r < x \leq s$.

Could someone verify my proof? Definition: Suppose $s \in \mathbb{R}$ and upper bounded $A \subset \mathbb{R}$. For any $x \in A$, we have $x \leq s$. For any $v$ such that $x \leq v$ for any $x$, we ...
1
vote
1answer
28 views

Understanding The Theorem “If there is a trail, then there is a path”

I am given the following theorem and proof: Statement Let $G=(V,E)$ be an undirected graph, $a,b\in V$, $a\neq b$. If there exists a trail(in $G$) from $a$ to $b$, then there is a path (in $G$) from ...
2
votes
1answer
57 views

Bijection on Preordered Sets Implies Homeomorphism

Prove that if $X$ and $Y$ are finite, then the "converse" of one of my other questions Homeomorphism on a Preordered Set is true: if $h: X \to Y$ is bijective and satisfies $\forall a,b \in X, ...
0
votes
3answers
45 views

Proof the the Arithmetic-Harmonic Mean is expressible as the Geometric Mean

We define the Arithmetic-Harmonic mean of $a,b \in \mathbb{R_+}$ such that \begin{gather*} a_{n+1} = \frac{1}{2}(a_n + b_n) \\ b_{n+1} = \frac{2a_{n}b_{n}}{a_{n} + b_{n}} \end{gather*} Let us also ...
0
votes
4answers
45 views

Prove: If $a$, $b$, and $c$ are consecutive integers such that $a< b < c $ then $a^3 + b^3 \neq c^3$.

Prove: If $a$, $b$, and $c$ are consecutive integers such that $a< b < c $ then $a^3 + b^3 \neq c^3$. My Attempt: I start with direct proof. Let $a,b,c$ be consecutive integers and ...
0
votes
0answers
33 views

Show that if $h$ is harmonic , then any mth order partial derivative of $h$ is a linear combination of certain partial derivatives

I want to solve the following exercise Show that if $h$ is harmonic , then any mth order partial derivative of $h$ is a linear combination of $\frac{\partial^{m}h}{\partial z^{m}}$ and ...
0
votes
2answers
39 views

Prove if A and B are n x n upper triangular matrices, so is AB

I'm trying to practice proofs for my linear algebra final and I've been stuck on this one for some time. I have $AB = [A\mathbf{b_1} \ A\mathbf{b_2} \ \dots \ A\mathbf{b_n}]$. I can show that ...
4
votes
1answer
43 views

Critique my elementary proof for a set bounded above

Let $A$ and $B$ be two non-empty subsets of $\mathbb{R}$ that are both bounded above. $(i)$Prove that $A ∪ B$ is bounded above and prove $(ii)$ that $\sup(A ∪ B) = \max(\sup(A),\sup(B))$. for ...
0
votes
1answer
29 views

Disprove for all integers $a$ and $b$ there exist integers $m$ and $n$ such that $a = m + n$ and $b = m − n$

Use Method of Contradiction to Disprove for all integers $a$ and $b$ there exist integers $m$ and $n$ such that $a = m + n$ and $b = m − n$ I get $\forall m,n \in \mathbb{Z},\exists ...
0
votes
1answer
25 views

Help understanding the proof for the infinity case of L' Hopital's Rule.

I am trying to understand the infinity case of L'Hopital's rule. I got this proof from Folland's Advanced Calculus. That is, suppose $f$ and $g$ are differentiable functions on $(a,b)$ and ...
0
votes
5answers
65 views

Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$

Let $a$ and $b$ be nonzero integers. Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$. Since this is a iff statement, I need to prove it both ways: $\Rightarrow$ If ...
0
votes
1answer
49 views

Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$

I want to prove this: $$(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$$ by every element on LHS(left hand side) is an element of RHS and vice versa. Does a counter example exist?
2
votes
1answer
65 views

Countability of a Set

Prove that a set $E$ is countable if and only if there is a surjection from $\mathbb{N}$ onto $E$. Suppose that $E$ is countable. Then there is a bijection from $\mathbb{N}$ to $E$ by definition of ...
4
votes
3answers
84 views

Prove even integer sum using induction

This is a homework problem, so please do not give the answer away. I must prove the following using mathematical induction: $\forall n\in\mathbb{Z^+},\;2+4+6+\cdots+2n=n^2+n.$ This is what I ...
-1
votes
1answer
74 views

How to solve a quintic congruence equation? [duplicate]

My textbook has this quadratic equation that I have to solve, any ideas how I could show that? $$15 | (21n^5+10n^3+14n),\;\forall n\in\mathbb{Z}$$
1
vote
1answer
50 views

How to prove that $\{a\} \times \{a\} = \{\{\{a\}\}\}$

I have to prove that $\{a\} \times \{a\} = \{\{\{a\}\}\}$. The cross product $\times$ between the sets $A$ and $B$ is defined as the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. ...
0
votes
0answers
5 views

How to generate a ploytope with a finite number of simplices

Well to solve this problem I wanted to show that a polytope with r- lineraly independent vertices is the finite union of r-simplices, but I am stuck in that because I dont know how to proceed and ...
1
vote
1answer
32 views

The number of vertices in a polytope is finite [duplicate]

I want to prove the following: Let $K$ be a convex polytope. Show that $K$ has a finite number of extreme points. I have seen the bound for the cardinality of the set of extreme points: $|E| \leq ...
0
votes
0answers
40 views

Homeomorphism on a Preordered Set

Prove that if $h$ is a homeomorphism from $(X,\mathscr{S})$ to $(Y,\mathscr{T})$, then $\forall a, b \in X \left( a \trianglelefteq_{\mathscr{S}} b \iff h(a) \trianglelefteq_{\mathscr{T}} h(b) ...
0
votes
1answer
32 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
0
votes
1answer
31 views

Show that the sum of the oscilations is less or equal to $f(b)-f(a)$

I want to show the following: Let $ f:[a,b]\to \mathbb{R}$ be an increasing function.If $ x_1,\ldots,x_k\in[a,b]$ are different, show that $$\displaystyle\sum_{i=1}^k O(f,x_i) < f(b) - f(a).$$ ...
0
votes
1answer
24 views

Optimal schedule for a set of jobs

Assume that yo have a set of jobs in which each has only a processing time that you need to minimize the sum of the completion (finish) times. Prove that your schedule is optimal. The wording throws ...
1
vote
2answers
20 views

DAG proof by numbering nodes

Prove that a directed graph is acyclic if and only if there is a way to number the nodes such that every edge goes from a lower number node to a higher numbered node. I know this is true and that ...
0
votes
0answers
18 views

Show that a compact polytope is the finite union of simplices.

This time I want to show that that a compact polytope is the finite union of simplices. I tried to show that if the polytope has r linearly independent vectors, then it has to be finite union of ...
1
vote
0answers
28 views

Equality for cartesian product

I need to get the cases when exactly (A $\cup$ B) × (C $\cup$ D) $\subset$ (AxC) $\cup$ (BxD) I have proved it everything that I did seems logical so I wanted to get a second opinion on my work here ...
0
votes
1answer
23 views

How to determine the sign of $q^{n+1} - q^n$ in different cases

Today in math class we saw how to determine whether a sequence is strictly increasing, strictly decreasing, or constant. I had to express $u_{n+1} - u_{n}$ for a geometric sequence $(u_{n})_{n \geq ...
2
votes
3answers
55 views

Proof using strong induction [duplicate]

I need to prove/show that $n^3 \leq 3^n$ for all natural numbers $n$ by strong induction. I have no clue where to begin!!!! :( I know how to do the beginning steps of showing that it's true for $k = ...
2
votes
2answers
34 views

Prove by Induction Summation

Prove by induction: Given that $f(x) = x^{-1}$, then the $k$-th derivative of $f$ is given by $f^{\langle k\rangle}(x) = (−1)^k \cdot k!\;x^{−(k+1)}$ for all $k ≥ 1$. How do I go about proving this? ...
0
votes
1answer
27 views

Prove $[(0,2)]\neq [(1,1)]$

I am working on this question For points $(a, b)$, $(c, d) \in \mathbb{R^2}$ define $(a, b) \simeq (c, d)$ to mean that $a^2 + b^2 = c^2 + d^2$, show $[(0,2)]\neq [(1,1)]$ . Proof: Let $(a,b) ...
0
votes
2answers
23 views

Inverse function of a set larger than original set

I'm trying to work on my proof writing ability and tried to prove the following theorem. let$ \: f:A\rightarrow B \; ,\: C\subseteq A$ Prove $C\subseteq f^{-1}[f(C)]$ let $x\in C$ so $\: f(x)\in ...
5
votes
2answers
174 views

What exactly is the difference between weak and strong induction?

I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching the $k^{th}$ rung of a ladder and ...
0
votes
1answer
27 views

How to prove $B-1$ and $B+1$ are always palindromes in base $B$?

While driving to work I thought about a rule, problem is I don't know how to prove it. The rule is: For any base $B$ there will always be two palindrome numbers $A$ and $C$, whose values are: ...
0
votes
2answers
64 views

Basis and vector spaces with change of coordinates

Let $B = \{v_1,...,v_n\}$ be a basis for a vector space $V$ and let $u_1,..., u_k \in V$. If $\{[u_1]_B,...,[u_k]_B\}$ is linearly independent in $\mathbb{R^n}$, then $\{u_1,...,u_k\}$ is linearly ...
1
vote
2answers
41 views

Prove or Disprove: Subspaces, and Bases

Prove or disprove: If U is a subspace of a finite dimensional vector space V and B = {v1, . . . ,vn} is a basis for V, then some subset of B is a basis for U. So far, I don't know where to start. I ...
0
votes
0answers
20 views

Identifying assumptions in a proof of a limit

I'm following a proof which starts with following inequalities: $$ dvP_{i}(v+dv) \geq {S_{i}(v+dv) - S_{i}(v)} \geq dvP_{i}(v)$$ where $v\in S$ and $v+dv \in S$ Diving by $dv$ and as $lim_{x \to 0}$ ...