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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3answers
48 views

Prove that $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, x_o + \varepsilon)$.

Let $x_0$ and $x$ be real numbers and let $\varepsilon$ be a real number with $\varepsilon \gt0$. Prove that $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, ...
0
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2answers
52 views

How do I prove that the complement of the closed interval $[a,b]$ is an open set.

How do I prove that the complement of the closed interval $[a,b]$ is an open set. I have a theorem that says an open set is a union of open intervals. Can I simply say the complement of the closed ...
-3
votes
1answer
26 views

Proof of positive semidefinite projection [closed]

How to show the sol. of $\min \limits_{X \in \mathbb{S}^+}||X-C||_F^2$ is $U \hat \Lambda U^T$ where $\hat \Lambda = diag(max(0,\lambda_1), ... , max(0,\lambda_N))$, $C = U\Lambda U^T$ and $\Lambda ...
1
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2answers
33 views

Induction proof verificiation

P(n) = in a line of n people show that somewhere in the line a woman is directly in front of a man. The first person will always be a woman and the last person in the line will always be a man I ...
3
votes
2answers
57 views

Functions and Sets

Robert, Susan, and Thomas are the sole contestants in a lottery in which two prizes will be awarded. Three tickets with their names on them are placed in a hat. The person whose name is on the first ...
1
vote
1answer
34 views

Trouble proving floor function is onto?

I'm trying to do a proof of a floor function being onto, but I'm not sure where to go from here. I don't want to ask the question outright because I want to figure it out myself, but I know that if ...
0
votes
1answer
73 views

Similar Matrices and Nullspace Proof

Prove that if A and B are similar matrices, the dim $Null(A)$ = dim $Null(B)$ I'm not really sure where to start for this problem. Any help would be appreciated. Thanks
2
votes
2answers
64 views

Prove that f'=f iff f is an exponential funtion

Written more formally, prove that $f' = f \iff \exists c \in \mathbb{R} : f = c * \exp$ In other words, I guess, it's enough to prove that $\exp$ and $f(x) = 0$ are the only functions that are equal ...
0
votes
2answers
62 views

Prove that $R- \{1,2\}$ is an open set

How would I show that the complement of the closed interval $[a,b]$ is an open set. My definition of an open set is: A subset $U$ of $R$ is called an open set if $U = \emptyset$ or if for each $x ...
0
votes
2answers
16 views

Prove an existential quantifier goal by assuming there exists an arbitrary value that makes the expression true.

I'm trying to prove the following: Suppose { A$_{i}$ | i $\in$ I } is an indexed family of sets and I $\neq$ $\emptyset$. Prove that $\cap$$_{i \in I}$A$_{i}$ $\in$ $\cap$$_{i \in ...
2
votes
2answers
46 views

Prove that $X\triangle\emptyset=X$

I'm working on my proofs involving sets, though this one is not a homework problem, so if you wish to provide your own example, so be it. I am working on exercise 3.3.14 (1) in Bloch's Proofs and ...
1
vote
2answers
33 views

Convergence in Complex Plane

Suppose that $z_n,z \in G = \mathbb{C} \setminus \{z:z \leq 0 \}$ and $z_n=r_ne^{i\theta_n}, z = re^{i\theta}$ where $- \pi < \theta, \theta_n < \pi$. Prove that if $z_n \to z$, then $\theta_n ...
2
votes
1answer
30 views

Show that if $A$ is any subset of a topological space $(X,T)$, then $Int(A) = \complement ( \overline {\complement (A)})$

Show that if $A$ is any subset of a topological space $(X,T)$, then $Int(A) = \complement ( \overline {\complement (A)})$ My reasoning went as follows: $\overline {\complement (A)} = \complement (A) ...
0
votes
1answer
33 views

Prove that in $\Bbb R$, $Int ([0,1]) = (0,1) $

Basically I need to show $Int([0,1]) = (0,1)$ meaning that I need to show that: $(0,1) = \bigcup_{a \in A}a$ Where for all $a \in A, a = (b,c)$ where $b,c$ real numbers such that $0 <b <c ...
2
votes
2answers
48 views

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge. My proof : We know that $\{a_n\}$ converge therefore : $$\lim_{n \to \infty} a_n = L$$ All $\epsilon>0$ exist $N \in \mathbb{N}$ so ...
0
votes
0answers
12 views

How can i explain the symmetry of the function in a more linguistic manner

To understand the convolution of these functions, please read the following wikipedia page I have the following expression $a = <f'(x),f''(x),\cdots>$ and $b = <g'(x),g''(x),\cdots >$ ...
0
votes
1answer
26 views

Proving $f(f^{-1}(D)) \subset D$

Suppose that $f:A \rightarrow B$ and let $D \subset B$. For proving $f(f^{-1}(D)) \subset D$: Let $x \in f(f^{-1}(D))$. Now $f(f^{-1}(D)) \in B$, so $x \in B$. Then $\exists y \in A$ such that $f(y) ...
1
vote
0answers
31 views

Probability and expectation

We are told that Alice and Bob each have a box that contains $n$ balls numbered $1,2,\ldots,n$. They pick balls from their boxes in a series of rounds as follows: In every round, Alice draws a random ...
2
votes
1answer
55 views

differentiable on $\Bbb R^{n}× \Bbb R^{n}$

Let $f : \Bbb R^{n} × \Bbb R^{n} → \Bbb R$ be defined by $f(x, y) = x·y$ , Show that $f$ is differentiable on $\Bbb R^{n}× \Bbb R^{n}$ and that $Df(a, b)(x, y) =b · x + a · y$ Here . denotes the dot ...
0
votes
1answer
23 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
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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
40 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
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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 ...
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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
35 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
40 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
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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
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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
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3answers
40 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
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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
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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
76 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
43 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
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1answer
34 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
37 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
38 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
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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
55 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, ...