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.
1
vote
0answers
73 views
Is this theorem proof correct?
I'm trying to prove this theorem:
Let $c_1$ and $c_2$ be real numbers with $c_2 \ne 0$. Suppose that $r^2 − c_1r − c_2 = 0$ has only one root $r_0$. A sequence $\{a_n\}$ is a solution of the ...
5
votes
2answers
71 views
Product of a family of spaces of countable tightness
I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem?
Finite family of compact spaces of countable ...
1
vote
1answer
39 views
Inverse implies surjection and follow-your-nose proofs
(I'm posting this question with my own answer, to show a nice calculational proof for one of the examples in Luke Palmer's blog post Follow Your Nose Proofs.)
In what follows, $A$ and $B$ are sets, ...
0
votes
0answers
18 views
Suggestions or comments for improving this proof
For a class paper I have written the following proof:
Given an array $a$ of size $n$ and $n = 2^{\lceil \log_2(l) \rceil}$, i.e. the next greater power of 2 of $l$. It follows that:
$$2^{\lceil ...
2
votes
3answers
119 views
Proving the so-called “Well Ordering Principle”
Is there anything wrong with the following proof?
Theorem. Every non-null subset $B \subset\mathbb{N}$ has a least member.
Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
3
votes
3answers
48 views
Prove the following is a tautology
I was trying to prove this statement is a tautology without using truth tables. Something doesn't add it here as I keep getting stuck. Take a look please!
For statements, P, Q and R prove that ...
1
vote
0answers
103 views
Proof Strategy for a Dynamical System of Points on the Plane
I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
0
votes
0answers
12 views
How to justify the equality $-\int_{0}^{2\pi}k^2v\ du=-\int_{0}^Lk^2ds$ in this proof?
Consider the curvature flow $$\frac{\partial F}{\partial t}=kN,$$ where $k$ is the curvature and $N$ the inner unit normal and $F:S^1\times [0, T)\rightarrow \mathbb R^2$ is a family of closed curves. ...
1
vote
1answer
40 views
Proving $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$.
Let $\mathscr F$ denote the set of all functions from $\{1,2,3\}$ to $\{1,2,3\}$.
Prove or disprove.
(a) $\forall f\in \mathscr F\exists g\in \mathscr F $ so that $g(f(1)) = 2$.
(b) $\forall f\in ...
1
vote
1answer
38 views
Is my proof showing $Q$ is non-measurable complete?
Is my proof valid or complete? if not, what is missing??
Define $R$ on $[0, 1)$. When $x,y\in
[0, 1)$, $xRy \iff y-x$ is a rational number. Then $R$ is an equivalence
relation. Let $Q$ be the set ...
0
votes
1answer
29 views
Bolzano-Weierstrass theorems question
Prove following theorem.
Theorem : If $x$ is a sequence of real numbers that is both bounded and monotone, then $x$ converges.
I know that $x$ is a sequence of real numbers that is both bounded and ...
4
votes
3answers
103 views
Proving or Disproving the Sum in a Set
I am doing review questions for an exam and I am completely stumped on this particular question:
Let A = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32}. Prove or disprove that if I select 10 distinct ...
1
vote
1answer
25 views
Finding the probability that X will be successful if its success is predicted
Consider an electronics company is planning to introduce a new
camera phone. The company commissions a marketing
report for each newproduct that predicts either the success
or the failure of the ...
11
votes
1answer
142 views
Is there such a thing as a mathematical thesaurus?
I want this for two reasons:
When writing proofs, I am constantly in need of synonyms of basic words like thus, there exists, for all, such as, contains, etc.
A lot of mathematical concepts have ...
1
vote
1answer
40 views
Proof of algorithm refinement
I recently had an interview in which I was asked to produce an algorithm to that computes the pairs of integers, from a list, that add up to a integer k.
I then had to increase the time efficiency of ...
3
votes
4answers
136 views
Proving that $T(n) = 3T\left(\frac n3\right) + \sqrt n = \Theta(n)$
Show that $T(n)$ is bounded both above and below by $n$ (abusing the Big O notation) for some positive constants $c_1$ and $c_2$:
$$
T(n) = 3T\left(\frac n3\right) + \sqrt n = \Theta(n)
$$
...
5
votes
5answers
92 views
Prove $(A \triangle B) \cap (B\triangle C) \cap (C\triangle A) = \emptyset$
This can be proved by assuming that there exists some $x \in (A \triangle B) \cap (B\triangle C) \cap (C\triangle A) $ and then deriving a contradiction by considering each of the cases that arise.
...
3
votes
0answers
39 views
(DFM) vs (DFS) spaces, Banach scales
I have already posted this question on MO:
http://mathoverflow.net/questions/126007/dfm-vs-dfs-spaces-banach-scales
However, not having received any feedback, I decided to repost it, since I have the ...
0
votes
0answers
28 views
Proof that a sequence of numbers $A_i$ is an infinite product of complex residues correct? [closed]
Is the above proof correct? Assume that theorem 2 is true without proof
Thanks
Edit: At the end it should be $2^n$, not $2n$
2
votes
1answer
19 views
Show that functions of one equation satisfy another
I am trying to show that if $A(x) $ and $ B(t)$ are a solution to
$$c A(x) + d B(t) = 0$$
where $c$ and $d$ are non-zero constants, then $A(x) $ and $ B(t)$ automatically satisfy
...
3
votes
4answers
70 views
Proof of $\sqrt{2^{2^k}} = 2^{2^{k-1}}$?
It's quite easy to observe that for $k \ge 0$:
$$
\begin{align}
2^{2^k} &= 4, 16, 256, 65536, \dots\\
\sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots
\end{align}
$$
More in general:
$$
\sqrt{2^{2^k}} ...
2
votes
2answers
49 views
For any normal subgroup $(aN)^n=(a^n) N$ holds
Prove this theorem
Let $G$ be a group and $N$ a normal subgroup of $G$. If $a \in G$ and $n \in Z$, then $(aN)^n = (a^n) N$.
I know I should prove this theorem in 3 cases where $n = 0$, $n>0$, ...
4
votes
4answers
84 views
Prove that $A \mathop \triangle C \subseteq (A \mathop \triangle B)\cup (B \mathop \triangle C)$
Suppose A, B, and C are sets, prove that $A \mathop \triangle C \subseteq (A \mathop \triangle B)\cup (B \mathop \triangle C)$
I'm just wondering if this proof is ok, or if I'm overlooking something, ...
1
vote
1answer
56 views
Proof of limit ratio theorem
My professor defines the Limit Ratio Theorem as follows:
Assume that $\displaystyle\lim_{n \mapsto \infty} \frac{f(n)}{g(n)}=c$, where $c$ is a constant or $\infty$.
If $0 \leq c < ...
1
vote
2answers
55 views
How to calculate this?$S=\sum\limits_{i=1}^\infty(-1)^i(i-1)!x^i$
How to calculate this equation $S=\sum\limits_{i=1}^\infty(-1)^i(i-1)!x^i$ ?
3
votes
1answer
85 views
Prove $\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le 2$
If $a,b,c,d > 0$ and $abcd = 1$, prove that the following inequality holds:
$$\frac{1}{bc + cd + da -1} + \frac{1}{ab + cd + da -1} + \frac{1}{ab + bc + da -1} + \frac{1}{ab + bc + cd -1} \le ...
2
votes
1answer
162 views
Error in Proof of Residues?
I wanted to prove that the function
$$F(z) = \frac{z-\sum_{j =2}^{n-1} z^j}{1-\sum_{k=1}^{n} z^k} $$
will only contain simple poles. Is the following proof correct?
Which implies that $z_o$ ...
1
vote
1answer
94 views
Using Cauchy's Inequality to prove a function's second derivative is zero
Let $f$ be an entire function (i.e. analytic everywhere, i.e. holomorphic) such that $\left\vert f(z) \right\vert \leq A \left\vert z \right\vert$, $\forall z \in \Bbb{C} $, where $A$ is a fixed ...
4
votes
0answers
108 views
extension of Cauchy's Integral formula
This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6.
Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
3
votes
4answers
105 views
Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
I want to prove by induction that $a^n \mid b^n$ implies that $a \mid b$ holds for all integers $n\geq 1$.
clearly for $n=1$ this is true, since if $a \mid b$, then $a \mid b$.
Suppose this is true ...
6
votes
3answers
98 views
Prove that if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$. [duplicate]
Prove that for any sets $A$ or $B$, if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$. ($\mathcal P$ is the power set.)
I'm having trouble ...
0
votes
1answer
30 views
Finding the pre-image $f^{-1}(T)$ for $T = [4, 9)$ and $f(x) = x^2$
This is a homework question. My class is titled "Formal Mathematical Reasoning and Writing" and we are using Lay's Analysis with an Introduction to Proof. My question comes from section 7: ...
2
votes
2answers
93 views
Prove that $A \subset B$ if and only if $A \cap B = A$
Prove that $A \subset B$ if and only if $A \cap B = A$
I'm having problems writing this proof using formal logic operators. I know the idea behind it: Since $A \subset B$, the only elements A and ...
5
votes
5answers
237 views
Minimum of $x^2 + 3x - 1$ on $[0,1]$ and $[-2,2]$
Consider the problem of finding the absolute minimum of the function $f : [0,1] \rightarrow \mathbb{R}$ that satisfies $f(x)=x^2 + 3x - 1$ everywhere.
Suppose we suspect, by graphical methods, that ...
5
votes
4answers
121 views
Prove that if $A \mathop \triangle B \subseteq A$ then $B\subseteq A$
I'm having trouble with the logic in this proof and was wondering if anyone could point me in the right direction (if I'm wrong)?
Prove that if $A\mathop\triangle B\subseteq A$ then $B\subseteq A$. ...
5
votes
5answers
114 views
Proving $n+3 \mid 3n^3-11n+48$
I'm really stuck while I'm trying to prove this statement:
$\forall n \in \mathbb{N},\quad (n+3) \mid (3n^3-11n+48)$.
I couldn't even how to start.
5
votes
1answer
69 views
If F→G is a consequence of F, then so is ¬G→¬F. A direct proof?
Homework question (introduction to logic):
"If $F \to G$ is a consequence of $\mathcal F$, then so is $\lnot G \to \lnot F$. We refer to this rule as $\to$-contrapositive. Verify this rule by giving ...
1
vote
2answers
120 views
Proof Help: Membership Table
I am new to proofs with membership tables and this is the last question I am posting.
I am trying to teach myself discrete math and am stuck on this:
Let $ A, B$ and $C$ be sets in the universal ...
0
votes
4answers
85 views
Proving by induction that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$
Need guidance on this proof by mathematical induction. I am new to this type of math and don't know how exactly to get started.
$$
1^3 + 2^3 + 3^3 + \ldots + n^3 = ...
0
votes
1answer
55 views
Big-O Big theta Big omega papers
I'm studying algorithms complexities by myself (my university didn't it to me) and I'd love if someone could help me in finding good resources to learn fundamental algorithms complexities proofing. ...
2
votes
3answers
106 views
Show that if $n$ and $k$ are integers with $1 ≤ k ≤ n$, then ${n\choose k} \le (n^k)/ 2^{k−1}$
I've looked everywhere but I've been unable to come up with a way to show that if $n$ and $k$ are both integers such that $1 ≤ k ≤ n$, then:
$${n \choose k} \le \frac{n^k}{2^{k−1}}$$
Thank you!
0
votes
2answers
74 views
Complex Numbers Proof
I have the statement, $|1 -zw^*|^2 - |z-w|^2 = (1 - |z|^2)(1-|w|)^2)$
Expanding $|1 -zw^*|^2$ gives me:
$1 - z^*w -zw^* - |z|^2$
Expanding $|z-w|^2$ gives:
$|z|^2 - zw^* - z^*w + |w|^2$
Subtracting ...
1
vote
2answers
88 views
Can anyone help me understand this Strong Mathematical Induction proof?
I'm not sure if we're allowed to post pictures but I thought it would be easier to read and I didn't see anything in the rules about it. It's question 1. Section 5.4
This question:
Here is the ...
1
vote
2answers
57 views
Probability Proof.
Write a proof to show that $\mathbb{P}(X_1 \mid X_3) + \mathbb{P}(X_2 \mid X_3) - \mathbb{P}(C_1 \cap X_2 \mid X_3) = \mathbb{P}(X_1 \cup X_2\mid X_3)$ labeling theorems used for each step.
My ...
15
votes
4answers
263 views
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.
Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Here's my idea:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
...
1
vote
0answers
36 views
Find the number of permutations in these words
Finding the number of permutations in these three words, am I doing this correctly?
a) CORRECT = $\frac{7!}{2!\cdot2!} = 1260$
b) COEFFICIENT = $\frac{11!}{2!\cdot2!\cdot2!\cdot2!} = 2494800$
c) ...
1
vote
1answer
18 views
How many permutations of this set can be made?
How many permutations of the set of seven letters (A,B,C,D,E,F,G) have the two vowels before the five consonants?
I'm wondering here if we use the set of 7! - 2! since they can only occupy the first ...
1
vote
3answers
118 views
Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$
How can we show that:
$$3n< n!$$
whenever $n$ is an integer such that $n \geq 7$ ?
I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
1
vote
3answers
86 views
How many different permutations of this set don't have vowels on the ends?
If we have the set of seven letters: (A,B,C,D,E,F,G) then how many permutations of these seven letters do not have vowels on the ends (that is, both the first and last letters are consonants)? I was ...
0
votes
0answers
32 views
I wanna prove if the composite are equal to each other
Given $f : \{0,1\}^n \to \{0,1\}^n$, define $f': \{0,1\}^{2n} \to \{0,1\}^{2n}$ as follows: for $x, r \in \{0,1\}^n$ define $f'(x \circ r) := f(x) \circ r$ (where $\circ$ denotes concatenation). Prove ...
