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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2answers
37 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
0
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0answers
6 views

Help in understanding proof to Darboux sum comparison lemma

I have some queries pertaining to the proof of the Darboux sum comparison lemma in the textbook, Advanced Calculus(Patrick Fitzpatrick). Kindly refer to lemma 7.10 on p 184 and its proof from pp 184 ...
0
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0answers
22 views
0
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2answers
26 views

Proof that minimum of sum of absolute differences is greater or equal of max value minus min value

Let's have an vector of natural numbers $[v_1, ..., v_N]$ my goal is to show that $$\sum_{i=1}^{N-1}|v_i - v_{i+1}| \ge v_{max} - v_{min}$$ where $v_{max} = \max_{i\in1...N}(v_i)$ and $v_{min} = ...
1
vote
1answer
53 views

Prove by induction that numbers of Fibonacci of the form $F_{3n}$ are even

I am not quite asking to solve this problem, but I am asking what they are asking me. This is the problem: The Fibonacci numbers $F_n$ for $n \in \mathbb{N}$ are defined by $F_0 = 0$, $F_1 = 1$, ...
1
vote
3answers
75 views

How to resolve $x \in A \wedge x \notin A $?

Let A and B be two sets. Then $A \setminus B = \{x: x\in A \wedge x\notin B\}$ $A \setminus B = \{x: x\in A \wedge x\notin A \cap B\}$ How can one prove that two logical statements are equal? ...
0
votes
2answers
42 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
1
vote
1answer
64 views

How do I prove that every open interval that contains $ \{ 1,2\}$ must also contain 1.5?

How do I prove that every open interval that contains $ \{ 1,2\}$ must also contain 1.5? $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, x_o + \varepsilon)$ We ...
0
votes
3answers
47 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
votes
2answers
49 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
24 views

Proof of positive semidefinite projection [on hold]

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
vote
2answers
31 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
52 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
31 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
33 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
63 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
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2answers
59 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 ...
0
votes
2answers
26 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
29 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
31 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
47 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
11 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
24 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
54 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
21 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
41 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
11 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
39 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 ...
0
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0answers
11 views

Proving function based on probability bound

I was initially working on a problem in which I was considering a set of positive integers $x_1,\ldots, x_l$ in $\{1,\ldots, n\}$ (defining the random variable $X= b_1 x_1+ \cdots + b_l x_l$, where ...
1
vote
1answer
29 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
80 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
30 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, ...
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votes
1answer
22 views

Proving a relation is reflexive (discrete) [closed]

I have been assigned a problem in discrete mathematics that says: Supposes R is a relation on N4 (first 4 natural numbers) such that R•R=R, prove R is reflexive. (• is the composition symbol). How ...
3
votes
0answers
34 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
53 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
2answers
46 views

Prove that if $r$ is irrational then $\sqrt [5]r$ is also irrational [closed]

If $r$ is an irrational number, then show that $\sqrt [5]r$ is also irrational. How to prove this by contradiction?
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1answer
59 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
35 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
39 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
33 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
28 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 ...
0
votes
1answer
54 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
39 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
38 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
67 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 ...