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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1answer
31 views

Show any open interval is a half open set?

How do I show that any open interval is an half open set and use this to conclude that any open set is also half open? I am in an introduction to proofs writing class. I have a feeling I need to use ...
2
votes
0answers
23 views

My proof that sum of convergent sequences converges to sum of limits

Does my proof appear correct? Also, do you like the notation? $\textbf{Theorem.}$ If $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are convergent real sequences, then $$ \lim_{n \to ...
-2
votes
0answers
38 views

Kid Genius Theorems. Where are the proofs? [on hold]

I'm looking for the proofs of the following theorems discovered by kids. I feel like there is tremendous value in being able to see a basic concept in a different light. ...
0
votes
1answer
9 views

Let X be a nonempty set. Let x∈X. Show that the collection 𝔗={U⊆X:U=∅ or x∈U} is a topology for X.

Let $X$ be a nonempty set. Let $x \in X$. Show that the collection $ \mathfrak T = \{ U \subseteq X : U = \emptyset$ or $ x \in U \}$ is a topology for X. I know I need to show that this ...
0
votes
2answers
29 views

What would the correct English description be for the difference for 1/3

If you measure a task & it takes 3 seconds, then the next time you do the same task, it takes you 1 second, is the difference 200% or 67%? Or would you say the difference is 200% because 3-1=2 ...
0
votes
1answer
35 views

Proof by contradiction or contrapositive sets help

so I'm having difficulties proving the following Theorem, through either proof by contradiction or contrapositive. Can someone please help me? The problem is as follows: Prove that for any two sets, ...
0
votes
0answers
25 views

Show that the interval $[0,2)$ is *H*-open but not *U*-open.

Show that the interval $[0,2)$ is H-open but not U-open. My definition of of H-open is a subset U of $\mathbf R$ is called an H-open set if $ U = \emptyset$ or if, for each $ x \in U$ , there is an ...
0
votes
1answer
24 views

Iterating proof step

Many books proves theorems by performing one proof step and using this step as a scheme they say by repeating this step $l$ times we prove that... I wonder whether there is some formal meta-theorem ...
2
votes
1answer
35 views

Prove that $n(r) < 2\pi \sqrt[3]{r^{2}}$

Suppose that $n(r)$ denotes the numbers of points with integer coordinates on a circle of radius $r > 1$. Prove that $$ n(r) < 2\pi \sqrt[3]{r^{2}} $$ What process would you use to resolve ...
-1
votes
1answer
19 views

Mathematical Induction Proof Question dealing with functions [on hold]

How would you use mathematical induction to prove: Let $f$ be a function of two positive integer variables with $f(1,1) = 2$ and $f(m + 1, n) = f(m,n) + 2(m + n)$ $f(m, n + 1) = f(m,n) + ...
0
votes
3answers
33 views

Mathematical Induction Proof Question dealing with integers

How would you use mathematical induction to prove that $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n \cdot (n + 1) \cdot (n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$ I tried proving the base ...
1
vote
2answers
64 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
votes
0answers
25 views
0
votes
2answers
32 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
55 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
78 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
43 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
66 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
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
votes
2answers
51 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 [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
vote
2answers
32 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
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
51 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
votes
2answers
61 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
27 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
32 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
25 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
22 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
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 ...
0
votes
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, ...
3
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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 ...
-2
votes
2answers
48 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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votes
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 ...