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

0
votes
1answer
24 views

Finite Sets Proof on Domains [duplicate]

Just wanted some help with this little proof.: Let X and Y be Finite Sets. Prove that |X^Y| = |X|^|Y|
0
votes
1answer
43 views

Simple Linear Algebra Proof - Determinants

Prove or disprove the following statement: If R is the RREF of A, then det A = det R. So far, I think that this is true, considering A and R are row equivalent, and that the determinant changes as ...
0
votes
0answers
13 views

Closure of a set with specified distance condition

Salam. I've presented the question and my thoughts on it. The question states: Let $S$ be a subset of $\Bbb R$ and $a \in \Bbb R$. Prove that $a \in \overline{S}$ if and only if for each positive ...
1
vote
1answer
42 views

How to proof: The set A × B is infinite if either A or B is infinite and the other set is not the empty set.

I'm having some difficulty trying to prove that. Suppse $A$ is an infinite set and $B$ is the non empty set, I thoght of showing that the cartesian product of both sets have the same cardinality as ...
1
vote
1answer
32 views

Converse of Borel-Lebesgue in $\mathbb R^n$

Question: If every open cover of a set $X \subset \mathbb R^n$ admits a finite subcover, then $X$ is compact. Note: Definition: $X$ is said to be a compact set is if $X$ is bounded and closed. ...
3
votes
3answers
81 views

Is there a set of $4$ vectors in $\mathbb{R}^3$, any $3$ of which form a linearly independent set?

I have an exercise in my last assignment for linear algebra: Is there a set of $4$ vectors in $\mathbb{R}^3$, any $3$ of which form a linearly independent set? Prove. My answer intuitively is ...
1
vote
1answer
31 views

Topology generated by the circles on the plane with their centers on a line

Question: Let $S$ be the collection of all circles on the plane which have their centers on the X-axis. If $S$ is a subbasis for a topology $T$ on $\Bbb R^2$, describe the open sets in $(\Bbb R^2, ...
3
votes
0answers
36 views

Much more topology…

These are very similar to my last two questions. I provide them with my thoughts so far: $(1)$ Let $S$ be the collection of all straight lines in the plane which are parallel to the x-axis. If $S$ is ...
2
votes
3answers
55 views

Show $P(1), P(2),…,P(99)$ true statements but $P(100)$ is false.

Provide a sequence of statements, $P(n),$ for $n\in \mathbb{N}$ such that $P(1), P(2),...,P(99)$ are all true but $P(100)$ is a false statement. My try: Let $n\in \mathbb{N}$ and let $0\notin ...
-2
votes
1answer
61 views

When $S$, the set of straight lines in $\mathbb{R}^2$, is a subbasis for a topology, what is the topology?

Salam. What are your thoughts on this? Again, please provide as much detail in your explanation as possible... Let $S$ be the collection of all straight lines in the plane $\Bbb R^2$. If $S$ is a ...
0
votes
2answers
40 views

mathematical induction to establish inequality

Studying for a test in discrete mathematics and I cannot seem to grasp the explanations in the textbook regarding these questions. Using mathematical induction, prove that $$2^n > n^2, \text{for ...
3
votes
1answer
40 views

Prove that $3\log n$ is $O(\exp(0.001n))$

First time posting here. Hi math stack-exchange community! I have a bonus question on my assignment and I am having trouble proving it. The main reason is that I am only limited to using the rules ...
2
votes
3answers
36 views

Proving that $2n^2 + n + 1 = O(n^2)$ and big O proofs in general

Alright so here's the thing, I'm in a class in Computer Science called Algorithm Analysis and it is required for me to learn Big O, Big Omega, etc. While I sort of understand what this is for, I still ...
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
41 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
78 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
63 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
44 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
38 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
41 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
31 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
38 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
31 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
31 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
23 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 ...