Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

0
votes
0answers
18 views

“Prove that ∂A ∪ Int(A) = Cl(A)” [duplicate]

The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. I have very little knowledge of set theory and proofs, so I'm not sure how to prove this. As ...
0
votes
2answers
30 views

Prove that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$

My textbook doesn't give me the fact that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$. We're asked to prove it. I'm given $\partial A = \mathrm{Cl}(A) − \mathrm{Int}(A)$. In a previous ...
2
votes
2answers
41 views

“Prove that ∂A is closed given ∂A = Cl(A) − Int(A)”

Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that ∂A = Cl(A) ∩ (Cl(X – A)). If that were the case, I could just state that definition ...
3
votes
2answers
77 views

Equivalent definition of ordinal sum

I'm trying to prove that two definitions of ordinal sum are equivalent. In the present question I want to prove that given $$\tag{1} \alpha+\beta:=\operatorname{ord}(\{0\}\times \alpha\cup \{1\}\...
1
vote
3answers
21 views

Fundemental Properties of Well-Ordering (Checking My Answers)

Give two different examples of well-ordered sets one of which is infinite. My Answer: $\mathbb{N}$ and $\left\{0\right\}$. Show that every subset of a well ordered set is a well ordered set with ...
0
votes
1answer
745 views

How to approach this proof on convergence of infinite product?

I have the following question I was given on a tutorial sheet to revise Infinite Series, and on it Infinite Products are introduced, as is the following question: Assume $\,b_{n}>0\,$ for all ...
0
votes
1answer
20 views

“Prove that Cl(ℚ) = ℝ in the standard topology on ℝ.” [duplicate]

I have very little knowledge of set theory and proof writing. This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. INFORMATION GIVEN Consider the ...
1
vote
1answer
34 views

How to show that $\sum \limits_{ \{x_1,x_2, \cdots , x_k \} \in S} \prod\limits_{i=1}^{k} x_i \equiv 0\ (\text {mod}\ p).$

Let $p$ be an odd prime. For each $1 \leq k \leq p-2$ consider the sets $S$ of all $k$-subsets of $\{1,2, \cdots, p-1 \}.$ Show that $$\sum \limits_{ \{x_1,x_2, \cdots , x_k \} \in S} \prod\limits_{i=...
0
votes
1answer
38 views

Types of Topologies: show how they compare to each other (finer, strictly finer, coarser…)

I have very little knowledge of set theory and proof writing. This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. DEFINITION Let $X$ be a set and ...
1
vote
1answer
39 views

Showing this Galois group is isomorphic to $\Bbb Z_4$ by examining the nature of its automorphisms.

Say we have the polynomial $x^6-1$, I want to find the Galois group of this over $\Bbb Q$. $x^6-1=(x-1)(x+1)(x^2-x+1)(x^2+x+1)$ And using the quadratic formula on the two quadratic polynomials we ...
0
votes
0answers
91 views

Proof by contradiction for search problem I believe is NP-hard

Intro Given that you solve the General Sudoku Problem(any puzzle validly solved). Is there an algorithm which has a hard-coded permutation $P$ and takes as input a valid 1st row of a Sudoku grid and ...
0
votes
1answer
33 views

“Show that a single-point set {n} is closed in the digital line topology if and only if n is even.”

I have very little knowledge of set theory and proof writing. This is a problem from Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa. DEFINITION: Digital line ...
0
votes
1answer
15 views

Existence and uniquenees to the equation

Prove that for every real number $x$, if $x \neq 0$ then there is a unique real number $y$ such that for every number $z$, $zy = \frac{z}{x}$. First I show that there is such that $y$, and second I ...
0
votes
2answers
18 views

Continuity on a discontinuous interval

I got a question which asked me to prove Intermediate Value Theorem for f:X->X where X = [0,1] U [2,3] and f is continuous I assumed that the function is either in [0,1] or [2,3] and so for the ...
1
vote
0answers
13 views

Prove that the positive integers $n$ and that cannot be written as a sum of $r$ consecutive positive integers are of the form $n=2^l$ for some $l>=0$.

Prove that the positive integers $n$ and that cannot be written as a sum of $r$ consecutive positive integers are of the form $n=2^l$ for some $l>=0$. Similar questions have been asked before but ...
0
votes
1answer
16 views

Bellman-Ford: Shortest path with $k$-edges

Let $G=(V,E)$ be a directed graph with weights $c:E\rightarrow\mathbb{R}$, $s,t\in V$ two nodes ($s\neq t$) and $k\geq 1$. How can I use the Bellman-Ford Algorithm to determine the shortest path ...
1
vote
1answer
45 views

Lawvere - Conceptual Mathematics - Criterion for decomposition of a map into two pieces

This is a problem from Lawvere's Conceptual Mathematics in a section talking about when we can find solutions to 'choice'/lifting diagrams. I have maps: $h\colon A \to C$, $g\colon B \to C$ with the ...
1
vote
1answer
49 views

The proof that the Mersenne number $M_{19}$ is prime.

Here is the hint to the proof given in the book: Using the following 2 theorems: 1-If $p$ is an odd prime, then any prime divisor of $M_{p}$ is of the form $2kp +1$. 2-If $p$ is an odd prime, then ...
2
votes
3answers
45 views

Determine which of the following are open sets in $\mathbb{R}l$. In each case, prove your assertion

I have very little knowledge of set theory and proof writing. This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. DEFINITION On $\mathbb{R}$, let $...
1
vote
1answer
44 views

Showing that a polynomial is surjective

How do I formally show that a polynomial, say $f(x)=x^3 - 5$, is surjective? For example, if $f(x)$ were a linear function such as $f(x) = 5x$, then I would simple need to show that for all $y$ in the ...
1
vote
1answer
30 views

Metric spaces bounded sets equivalence

Let X be a a non empty metric space and suppose A $\subseteq$ X. Then A is bounded if $\exists$ $z$ $\exists$ $r>0$ so that $A \subseteq B(z,r)$. WTS The following are equivalent: 1) A is ...
0
votes
0answers
29 views

Is it bad math style to use one letter for indexing two objects? [duplicate]

Is $$\sum_{i\in I_1} a_i + \sum_{i \in I_2} b_i$$ or $$\prod_{i\in I_1} a_i \cdot \prod_{i \in I_2} b_i$$ confusing or bad math? Should I rather use $j$ for indexing the $b$'s?
2
votes
5answers
79 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
0
votes
1answer
24 views

Can a well-ordered set isomorphic to a proper subset of itself?

Can a well-ordered set isomorphic to a proper subset of itself? Give an example or disprove it. My answer: Yes We can. Consider $2\mathbb{N}\subseteq\mathbb{N}$. Let $f$ be the function from $\mathbb{...
4
votes
1answer
77 views

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...
1
vote
0answers
40 views

Domain co-domain, proof check

Let $f:A\rightarrow\mathbb{R},g:B\rightarrow\mathbb{R},g \circ f:C\rightarrow\mathbb{R}$ Then we have: 1.$C\subseteq A$ 2.$f(C)\subseteq B$ Proof. $\forall c \in C,g \circ f$ is ...
0
votes
1answer
599 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection $\...
12
votes
5answers
344 views

Interesting inequality $\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[2m+1]{\frac{1+x^{2m+1}}{2}}$

Prove that $$\frac{x^{m+1}+1}{x^m+1} \ge \sqrt[k]{\frac{1+x^k}{2}}$$ for all real $x \ge 1$ and for all positive integers $m$ and $k \le 2m+1$. My work. If $k \le 2m+1$ then $$\sqrt[k]{\frac{1+x^k}{...
3
votes
2answers
925 views

Contrapositive of the statement with quantifiers

$\forall x$, $2 |x \implies x^2 = 4$ False statement but lets go with it Find the contrapositive: Would it be, $\forall x$ $x^2 \ne 4 \implies 2 \not | x$ OR $\exists x$ $x^2 \ne 4 \implies 2 \not |...
5
votes
1answer
37 views

Alternately Multiplying and Dividing Within Pascal's Triangle

I recently noticed that, if you take the $(2n)$th row of Pascal's triangle, and alternately multiply and divide the numbers that crop up, almost everything cancels. For example, the 4th row (1 4 6 4 1)...
1
vote
2answers
32 views

Prove using induction $\prod_{i=1}^n (1-2^{-i}) ≥ \frac{1}{4} + 2^{-(n+1)}$ for all n∈N

Prove using induction $$\prod_{i=1}^n (1-2^{-i}) ≥ \frac{1}{4} + 2^{-(n+1)}$$ for all n∈N My answer so far: base case: n = 1. P(1) = $$(1-2^{-1}) = \frac{1}{2} ≥ \frac{1}{4} + 2^{-(1+1)} $$ so it ...
0
votes
1answer
20 views

If $D$, $E$, $F$ are feet of perpendiculars from a point to the sides of $\triangle ABC$, then $BD^2-DC^2+CE^2-EA^2+AF^2-FB^2=0$

The actual question is If from a point $O$, segments $OD$, $OE$, $OF$ are drawn perpendicular to the sides $BC$, $CA$, $AB$, respectively, of $\triangle ABC$, then prove that: $$BD^2-DC^2+CE^2-...
0
votes
2answers
25 views

Angle bisector related problem

In $\Delta ABC$, $BE$ and $CF$ are the angular bisectors of $\angle B$ and $\angle C$ meeting at $I$. Prove that $AF/FI=AC/CI$ I have tried this question for hours but i am unable to hit the nut I ...
0
votes
0answers
24 views

Find invalid values of $Y$

Let $$Y = (\textbf{a})\cdot x_1 + (\textbf{a}+1)\cdot x_2 + (\textbf{a}+2)\cdot x_3 + (\textbf{a}+3)\cdot x_4 + . . . + (\textbf{a}+n)\cdot x_n$$ Where $x_1,x_2,x_3.... x_n$ are all positive integers ...
0
votes
1answer
43 views

Show there can't be two real and distinct roots of polynomial $f(x)=x^3-3x+k$ in $(0,1)$, for any value of k.

I have two proofs here one which I did and the other was given in book. Is one better than the other? I am asking for in an exam setting which proof makes a better solution(as in fetch more marks). ...
-7
votes
1answer
78 views

Philosophy of Math- Does Maths exist and what branches of maths support this [closed]

Clarifying Question What proof does mathematics offer that it is a universal rather than general/specific language? For context, please see the original message below. Original Message I want to ...
1
vote
5answers
1k views

Assume that 495 divides the integer $\overline{273x49y5}$ where $x,y \in \{0,1,2…9\}$. Find $x$ and $y$.

So, I know that $495 = 5\times 9\times 11$. So then, if that's the case, then the number $\overline{273x49y5}$ must be divisible by $495$ if and only if it is divisible by $5$ and $9$ and $11$. Then, ...
2
votes
1answer
50 views

Proving Differentiability of a Function (Spivak)

The following is an exercise from Calculus on Manifolds by Spivak. Let $g$ be a continuous, real-valued function on the unit circle $\{x\in\mathbb{R}^2:\|x\|=1\}$, such that $g(-x)=-g(x)$. Now ...
3
votes
3answers
48 views

Easier method of finding the equation of the circle circumscribing the triangle formed by 3 lines?

The equation of the circle circumscribing the triangle formed by the lines $y = 0, y = x$ and $2x + 3y = 10$ is? I know this can be done by solving two equations at a time and finding the vertex. ...
0
votes
1answer
24 views

How to show this set is irreducible

Proposition. Let $X$ be a topological space and $U\subseteq X$ an open set of $X$. If $Z$ is a closed irreducible set of $X$ that meets $U$ then $Z\cap U$ is a closed irreducible set of $U$. Note. ...
5
votes
1answer
55 views

Taking a natural number different from zero as a base case in an inductive proof

Principle of mathematical induction states that if a subset $S$ of a successor set $\omega$ is also a successor set, then $S=\omega$. In primitive terms, it is formulated as: if $S \subset \omega$, ...
0
votes
2answers
36 views

Number theory modulo problem [duplicate]

For a prime $p$, if $x^p \equiv y^p \pmod p $ Then prove that; $x^p \equiv y^p(\bmod p^2) $ Firstly I am confused where to approach to prove this question ,though i know several propositions and ...
0
votes
3answers
63 views

AM-GM Inequality Involving Squares and Proof

Prove: $$(a^2 + b^2 + c^2)/3 \geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \leq ((a + b + c)/3)^2$$ for all $a, b, c \geq 0.$ The problem wants me to find which inequality is correct and then ...
0
votes
1answer
39 views

To find hottest point on probe surface

A space probe in form of ellipsoid $4x^2+y^2+4z^2=16$ enters earth atmosphere and its surface begins to heat. After one hour, temperature at point $(x,y,z)$ on probe surface is given by $$T(x,y,z) = ...
1
vote
6answers
4k views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
0
votes
0answers
39 views

Need a logical proof of [(c>0) and (|a|<c)] implies [(-c<a) and (a<c)]

I need a logical proof of the elementary statement about real numbers using order and field axioms $((c>0)\wedge(|a|<c))\Rightarrow((-c<a)\wedge(a<c))$
1
vote
2answers
37 views

Polish spaces are continuous images of the Baire space

I'm having some troubles understanding the proof of Theorem 7.9 (pag. 39) in Kechris' "Classical Descriptive Set Theory": There are two points of the proof proposed that I don't quite understand. ...
13
votes
3answers
20k views

Show that the eigenvalues of a unitary matrix have modulus $1$

Show that the eigenvalues of a unitary matrix have modulus $1$. I know that a unitary matrix can be defined as a square complex matrix $A$, such that $$AA^*=A^*A=I$$ where $A^*$ is the conjugate ...
1
vote
1answer
52 views

Proving results on quadratic equations [duplicate]

Let $p \in \mathbb Z [x]$ be a monic quadratic polynomial. Show that for $n \in \mathbb Z$ there exists $k \in \mathbb Z$ such that $$p(n)p(n+1)=p(k)$$ I came across this problem in an olympiad book. ...
2
votes
1answer
43 views

ODE: $\dot{x}=f(t)x^2,x(0)=1$ solution on $\mathbb{R}\Leftrightarrow\int_{0}^{t}f(s)ds<1\forall t\in\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function and $\phi$ a solution on its maximal interval of existence for the initial value problem $$ \dot{x}=f(t)x^2\hspace{0.3cm},\hspace{0.3cm}...