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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7
votes
1answer
264 views

How to write well in analysis (calculus)?

This is kind of a subjective question, I know; often I find myself failing exams and homeworks because of the way i write down proofs. Either I don't know how to start, or somehow the main point of ...
2
votes
2answers
439 views

Infinum & Supremum: An Analysis on Relatedness

$\require{color}$ Question: I need some help in proving that $\color{green}{\text{if $k\geq 0$, then $\sup (kS) = k\sup(S)$ and $\inf (kS) = k\inf (S)$}}$, and also that $\color{red}{\text{if ...
1
vote
0answers
74 views

Stylish Academic Writing [duplicate]

I don't know that this question belongs here, but I'd like to know of any references out there anyone here might recommend for writers of mathematical ideas, be it a book, an article, a dissertation, ...
3
votes
2answers
100 views

Prove that $n + 2$ is odd where $n = 2k+1$ for some integer $k$

I am attempting to learn about mathematical proofs on my own and this is where I've started. I think I can prove this by induction. Something like: $n = 2k+1$ is odd by definition $n = 2k+1 + 2$ ...
2
votes
1answer
81 views

Proving a set is a metric space

I am currently doing questions from Kaplansky's Set Theory And Metric Spaces. I come to seek validation on my answers because my book does not have an answer key. I am looking on ways to strengthen my ...
2
votes
0answers
47 views

On the proof that the inverse value set of a regular value is a submanifold

I have a doubt on the proof of the following, well-known theorem: Let $f:M^m\rightarrow N^n$ ($m\geq n$) be a $C$ map, $r\geq 1$.If $y\in f(M)$ is a regular value, then $f^{-1}(y)$ is a $C$ ...
2
votes
0answers
38 views

Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
24
votes
6answers
765 views

Level of Rigor in Mathematical Physics

I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
0
votes
2answers
95 views

Proof by contradiction and division by $0$

Proof by contradiction is based on the fact that if, as a consequence of a statement's truth, we reach a contradiction, then that statement must be false, since contradictions do not exist in ...
1
vote
2answers
90 views

For h $\in \mathbb R$ and $h \gt -1$ and n $\in \mathbb N$: prove $1 + n \cdot h \le (1 + h)^n$

I try to prove this using contradiction, but something tells me this is not a valid proof. Suppose it's not true, and suppose $h \lt 0$. Then we get that $(1 + h)^n \gt 1$ and $1 + n \cdot h \lt 1$ ...
2
votes
1answer
107 views

Problem of proofs

I've been away from math for a long time ,and while I was trying to relearn it using Courant and Fritz 's booknon calculus,I loved the explanations but I couldn't solve any exercices(they're almost ...
0
votes
1answer
113 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
1
vote
3answers
82 views

Can we give an algorithm to prove a statement?

Can we prove a statement by providing an algorithm that is true for all conditions of the statement? Or do we need to prove the validity of the algorithm too? As an example, suppose we need to prove ...
1
vote
2answers
60 views

Exercise 1(d) from Courant

I'm having trouble understanding this "hint" in the back of (the first volume of) Courant's Differential and Integral Calculus text, which I'm just starting: One of the "challenging" Chapter 1 ...
2
votes
1answer
65 views

Are these two proofs regarding coloring valid and complete?

Question #1) Prove or disprove: If G is a graph and for every vertex $v \in V(G), \chi (G-v) < \chi (G)$, then for every subgraph H such that $H \neq G, \chi(H) < \chi(G)$. Question #2) Prove ...
1
vote
1answer
48 views

Elementary proof for $-v \leq u \leq v$ iif $|u| \leq v$

I'm having difficulties with writing proofs, probably because I've just started the subject. And i really would like to avoid looking at the answers and solve it as best as I can myself. Now I'm ...
1
vote
1answer
52 views

If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
3
votes
3answers
1k views

Real function continuous on closed interval implies it is bounded - over-simple proof??

So here is my proof, which after looking up others seems to be too simple, or not rigourous enough, though I don't see why (hence I am asking!): We take the contrapositive, and so prove that if $f$ ...
4
votes
4answers
295 views

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a ...
2
votes
1answer
90 views

Help with understanding a proof that $f$ is bounded on $[a,b]$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ ...
1
vote
2answers
67 views

Suppose $f: A\rightarrow B,g:B\rightarrow C, h:B\rightarrow C$. Pro that If $f$ is onto $B$ and $g\circ f$= $h\circ f$, then $g=h$

Suppose that $f: A\rightarrow B,g:B\rightarrow C, h:B\rightarrow C$. Prove that If $f$ is onto $B$ and $g\circ f$= $h\circ f$, then $g=h$ Could anyone please give some guideline on how to solve ...
4
votes
5answers
287 views

Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?

Suppose I have a rational number $a$ and $a^2 < 2$. Can I find another rational number $B$ such that $a^2<B^2<2$? Based on the answer to this question, I thought of doing the following: ...
3
votes
1answer
30 views

What should be proven when your claim ends with a clause for a particular case?

My question is not about how to prove the following proof, as I already know how to do so. What I am more concerned about is the type of claim that is being made, and what am I really being asked to ...
1
vote
1answer
606 views

Need proofread to show hermitian matrix has only real eigenvalues

Prove Hermitian matrix has only real eigenvalues $$\textbf{A}^\ast = \textbf{A}$$ Proof Let eigenvalue $\lambda \neq \vec{0}$ such as $$\textbf{A}\vec{v} = \lambda\vec{v}$$ $$\Rightarrow ...
2
votes
0answers
56 views

Is my proof correct? (Also formally)

Hello dear community! I just worked on a problem in my discrete mathematics text book and wondered if my approach to a specific exercise is correct. There are no solutions to it, that's the reason I ...
1
vote
1answer
809 views

Solving graph theory proofs

I am trying to study for an exam on graph theory and I have a few questions. How would you start a proof? For example, when I see a problem like this: Let G be a graph with n vertices where every ...
3
votes
4answers
560 views

How to prove Disjunction Elimination rule of inference

I've looked at the tableau proofs of many rules of inference (double-negation, disjunction is commutative, modus tollendo ponens, and others), and they all seem to use the so-called "or-elimination" ...
0
votes
1answer
77 views

how to prove a result

I was studying about graph operation on wiki. Corona product of graphs $G_1$ and $G_2$, is the graph which is the disjoint union of one copy of $G_1$ and $|V_1|$ copies of $G_2$ ($|V_1|$ is the number ...
1
vote
3answers
103 views

What is a self-contained proof?

For this question I am required to give a self-contained proof of a statement, but I am not sure what a "self-contained proof" is.
1
vote
1answer
102 views

Prove correctness for this lcm iterative program

Studying for finals, trying to solve this problem: Given positive integers $n$ and $m$, we say that $m$ is a multiple of $n$ iff there is some $k \in N$ such that $m = kn$. For positive ...
6
votes
4answers
314 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For ...
2
votes
2answers
251 views

Nested roots sequence, how to prove it's monotone and bounded?

Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$ Here's what I did: Plugging in some values makes it seem as if the sequence is increasing. I ...
0
votes
1answer
85 views

Simple Projection Proof

Let $V = U \oplus W$, and define $P_{U,W} \in L(V)$ where $P_{U,W}$ denotes the projection onto $U$ with null space $W$. I am trying to verify three properties and would like some feedback and help. ...
5
votes
1answer
96 views

Help to understand the proof of $ \lim \limits_{x\to 0^+} f \left(\frac{1}{x}\right)=\lim \limits_{x\to \infty}f(x)$

The following is an answer to the proof of $$ \lim \limits_{x\to 0^+}f\left( \frac{1}{x} \right)=\lim \limits_{x\to \infty}f(x)$$ If $l=\lim \limits_{x\to \infty}f(x)$, then for every ...
1
vote
4answers
260 views

Parallel Lines Proof

How do you prove that two parallel lines never cross? By definition this is implied, but how do you prove it for any pair of parallel lines? In other words, how do prove that 2 parallel lines will ...
6
votes
3answers
171 views

Let $f:A\rightarrow B$. Prove that if $ X\subseteq A$, and $f$ is one to one, then $f(A)-f(X) \subseteq f(A-X)$.

Let $f:A\rightarrow B$. Prove that if $ X\subseteq A$, and $f$ is one to one, then $f(A)-f(X) \subseteq f(A-X)$. Could anyone please guide me through this problem? I got stuck and don't know if what ...
1
vote
3answers
630 views

Prove that $P(X)$ has exactly $\binom nk$ subsets of $X$ of $k$ elements each.

Let set $X$ consist of $n$ members. $P(X)$ is power set of $X$. Prove that set $P(X)$ has exactly $$\binom nk = \frac{n!}{k!(n-k)!}$$ subsets of $X$ of $k$ elements each. Hence, show that $P(X)$ ...
3
votes
3answers
164 views

Is it proof by contradiction?

In a proof of the statement "A closed ball $\bar{B}\left(a,r\right)$ is a closed set", author first took complement of $\bar{B}\left(a,r\right)$ and proved it an open ball. I was wondering whether ...
1
vote
1answer
44 views

How to find the $l$ in $-\epsilon + 1 < l < \epsilon + 1$?

Suppose we have: $$ \epsilon > 0 $$ and $$ -\epsilon + 1 < l < \epsilon +1 $$ I think I have enough information to show that $l=1$ but I don't know how to formally show that this is true. ...
6
votes
4answers
844 views

What character can replace word “let” in proofs?

For example, suppose I have a line of a proof introducing new “variable” $x$: $$\textrm{Let}\:\:x\in f(y)$$ I am looking for ways to express the word “let” in this context and ...
0
votes
2answers
89 views

Backward induction (Tao Analysis vol. 1).

Exercise 2.2.6: Let $n$ be a natural number, and let $P(m)$ be a property pertaining to natural numbers such that whenever $P(m+1)$ is true, then $P(m)$ is also true. Suppose that also $P(n)$ is ...
0
votes
1answer
31 views

Is this graph transitive?

I have a graph $G = (A,B)$ which is transitive when: $(a,b) ∈ B ∧ (b,c) ∈ B → (a,c) ∈ B$. How can I prove that $G$ is transitive iff it's acyclic?
0
votes
1answer
17 views

Question about graphs and relations

If I have a directed graph $G = (V,E)$, let the relation $R$= {$(a,b)$ | $a$ has a directed path to $b$} be a relation over $V$. How can I prove that $R$ is an equivalence relation, partial order, ...
1
vote
1answer
326 views

Proving that a square can be divided into $n$ smaller squares for $n \ge 6$

I'm trying to prove that for all natural numbers $n \ge 6$, a square can be divided into $n$ smaller squares. The smaller squares do not need to be of the same size. So for induction, the base case ...
0
votes
0answers
65 views

Mathematical induction, equivalence of formulations, check my proof please.

I've starting going through Tao's Analysis I and in the first chapter there is an exercise about proving the equivalence of weak and strong induction. In the text, the principle of induction is an ...
0
votes
1answer
40 views

How to use first-order logic with both constants and predicates?

I'm trying to create a first order logic statement, but I have constants in addition to predicates. Predicates: $Time(a)$, which states that a is a time; $After(a,b)$, which states that $a$ is ...
0
votes
1answer
112 views

Problem with proving Catalan number

This is how my professor derived it: Taking the case of all valid arrangements of $n$ '(' and $n$ ')', he says that for every invalid arrangement, there will be a ')' at some $k^{th}$ position where ...
2
votes
1answer
448 views

Help understanding the proof of Lame's Theorem.

I think Lamé's Theorem is beautiful and really want to understand the proof. I am new to proofs, but after reading over the proof of Lamé's Theorem (and failing to understand it), I feel that I am ...
0
votes
2answers
138 views

Stuck on question regarding Cantor's theorem and sets

I'm trying to prove that a set of all sets does not exist, meaning that the following does not exist: $$ D = \{ S \mid S \text{ is a set} \} $$ I can use Cantor's Theorem and the proof of ...
0
votes
2answers
37 views

How $\delta_1$ and $\delta_2$ for two different limits at $a$ can be read as $\delta=\text{min}(\delta_1,\delta_2)$?

I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous ...