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.

learn more… | top users | synonyms

3
votes
1answer
48 views

Is Alfred Tarski's Introduction to Logic still helpful for self study?

I am trying to setup a self study path to enhance my knowledge of mathematical logic. I haven't taken a logic course for a few years and my confidence on mathematical proofs is unnerving. I am ...
0
votes
1answer
30 views

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$.

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$. I know this is a true statement. I am trying to prove if because I would also ...
0
votes
1answer
64 views

Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$

I want to prove this: $$(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$$ by every element on LHS(left hand side) is an element of RHS and vice versa. Does a counter example exist?
2
votes
2answers
173 views

Prove without Parallel Postulate

Let $x$ and $y$ be parallel lines where $x\neq y$. How do I prove that $y$ is in one of the $1/2$ planes , let's call it $H$ of $x$ ? How to prove that one of $1/2$ planes of $y$ is contained in ...
1
vote
1answer
33 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
0
votes
1answer
64 views

Seemingly simple logic question

I found this pleasant textbook on Proof Theory online and free: Introduction to Proofs, an Inquiry-Based approach To quote (page 9): 2.26 DEFINITION. A sequence $\langle x_0,x_1, . . . ...
1
vote
2answers
30 views

Logically Equivalance - Proofs

In terms of logical statements, is ($\exists$n $\in$ N)($\forall$ x $\in$ A)(nx >= 1) equal to ($\forall$x $\in$ A)($\exists$ n $\in$ N)(nx >= 1)? Also consider the following statements $\forall x ...
2
votes
2answers
32 views

Logical Statements - Proofs

Let $f$ be a real-valued function (a function with target space the set of reals). Let $P(x, M)$ stand for $|f(x)| \leq M $, let $N$ be the set of positive real numbers, and let $\mathbb{R}$ be the ...
0
votes
2answers
40 views

Is a proof by counterexample considered a proof by contradiction?

My question is already in the title. Let us look at some example. I would like to prove that a game $G(n,m,u)$ does not have a pure Nash equilibrium (PNE), for example. I did it like this: Suppose ...
39
votes
2answers
932 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
2
votes
4answers
72 views

Non-standard model of $Th(\mathbb{R})$ with the same cardinality of $\mathbb{R}$

Let $\mathfrak{R}= ⟨\mathbb{R},<,+,-,\cdot,0,1⟩$ be the standard model of $Th(\mathbb{R})$ in the language of ordered fields. I need to show that there exists a (non standard) model of ...
0
votes
4answers
43 views

Logic, writing proof

i)Suppose that $x$ and $y$ are real numbers. Prove that if $x\neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$ ii)Suppose that $x$ and $y$ are real numbers. Prove that if $x^2y=2x+y$, ...
1
vote
0answers
20 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
0
votes
0answers
35 views

Principle of infinite descent

I'm trying to prove the following: "It is not possible to have a sequence of natural numbers which is in infinite descent. (Hint: assume for sake of contradiction that you can find a sequence of ...
2
votes
2answers
53 views

Number Theory- Mathematical Proof

I am trying to prove a theorem in my textbook using another theorem. What I need to show that if a,b, and c are positive integers, show that the least positive integer linear combination of a and b ...
0
votes
2answers
24 views

Prove that for all $x$, $y$ in $\mathbb{R}$ there exist $z$, $g$ such that $x = z + g$, $y = z - g$

if I want to prove the following: $\forall x, y \in \mathbb{R}\,\,\,\exists\,\,z, g : x = z + g, y = z - g$ Can the resolution of the following system act as a proof: $\begin{cases} x = z + g\\ y ...
3
votes
2answers
25 views

Check proof of some simple inequality

Can you check please my proof of this inequality? It's all right?
0
votes
0answers
21 views

Prove equality of Fibonacci sequence

Let $u(n)$ — the Fibonacci sequence. Prove that $$u(1)^3+...+u(n)^3=\frac{ 1 }{ 10 } \left[ u(3n+2)+(-1)^{n+1}6u(n-1)+5 \right].$$ I suppose we need prove that equality by iduction in $n$. ...
1
vote
0answers
12 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
3
votes
3answers
76 views

Prove $\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$ using $\delta-\epsilon$ definition of limit

I need to prove that the $$\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$$ using $\delta-\epsilon$ definition of limit. Now, I have started with a discussion, saying that what we want is that if $\left| x - ...
0
votes
1answer
35 views

Proving an Iff Statement

Suppose we had a function defined over the complex numbers: $ f(x)= \begin{cases} 1&\text{if } x\in\mathbb{R}\\ 0&\text{if } x\not\in\mathbb{R} \end{cases} $ And we are asked to prove that ...
2
votes
1answer
24 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
3
votes
3answers
61 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [on hold]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
6
votes
3answers
422 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
0
votes
2answers
47 views

On proving $(f^{-1})'(b) = \frac{1}{f'(a)}. $ where $b = f(a)$.

Could somebody kindly provide a proof or a reference to a proof of this fact: Let $ I $ be an open interval, and suppose that $ f: I \to \mathbb{R} $ is one-to-one and continuous on $ I $. If $ f ...
4
votes
2answers
89 views

Proof of determinant formula

I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my ...
0
votes
2answers
38 views

Proof in set theory

Let $A,B,C$ -- subsets in some fixed set. Prove that $A \cap B \subseteq C$ iff $A \subseteq \overline{B} \cup C$. Have no ideas how to prove this. On the language of definitions we have $$x ...
1
vote
2answers
62 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
1
vote
3answers
99 views

Prove that there are infinity many tautologies.

For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by ...
2
votes
4answers
222 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
0
votes
0answers
10 views

Theorem implication/equivalence transitiveness in demonstrations

Suppose having three theorems $A, B, C$ that it's necessary to show being equivalent and having the following hypothesis: We know that $A \Leftrightarrow B$ and $B \Leftrightarrow C$. It would ...
1
vote
2answers
80 views

Prove that $\sin^{2}{\theta} + \cos^{2}{\theta} = 1.$

I believe that I have been able to prove that Prove $\sin^{2}{\theta} + \cos^{2}{\theta} = 1, \forall \theta,$ but I would like to ask if my proof is correct / valid.
1
vote
2answers
323 views

Proof that arithmetic and geometric mean converge

I need some help with understanding a part of this proof and also writing it up correctly. Given $a_n\geq a_{n+1}\geq b_{n+1} \geq b_n$ with $a_1=a$ and $b_1=b$. I am also given that ...
0
votes
1answer
55 views

Some proofs regarding Stirling numbers

I would like you to help me to prove two proofs correlated with Stirling numbers (the first one includes Stirling numbers of the second kind and the second one I guess Stirling numbers of the second ...
1
vote
2answers
29 views

Path and cycle length proof

How can we prove this proposition : Every graph G contains a path of length $ \delta(G)$ and a cycle of length at least $ \delta(G) + 1$, provided that $\delta(G) \geq 2 $. ($ \delta(G)$ is the ...
0
votes
0answers
24 views

Comparison of books that teach proof techniques

I have to take discrete math and want to learn proof techniques both to get ahead in it as well as open up the possibility of understanding higher math. I've seen several books recommended such as How ...
1
vote
1answer
60 views

prove that $p(n) := n^2 + n + c$ is not prime

The question is in MIT Mathematics for CS assignments but unfortunately there is no solutions. -> I do understand that it is false if we use $n = c$ or $n = c-1$ but cannot formally write it as ...
3
votes
4answers
82 views

Prove every integer is of the form $5k+r$ with $0\le r<5$

I have came across this question from my text book: Prove or disprove: any integer $n$ is of the form: $5k$, $5k + 1$, $5k + 2$, $5k + 3$ or $5k + 4$ for some integer $k$. I'm not sure what would be ...
41
votes
8answers
4k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
0
votes
1answer
27 views

Primary decomposition of modules - uniqueness proof

Let $M$ be $A$-module, $A$ commutative ring, and $N$ submodule and let $$N=Q_1\cap\dots\cap Q_r=Q'_1\cap \dots \cap Q'_s$$ be reduced primary decompositions of $N$. Then $r=s$. The set of primes ...
5
votes
4answers
73 views

Show $x^n \geq x_1^n+x_2^n+\ldots+x_k^n \Bigg\vert x_1+x_2+\ldots+x_k = x, x \geq 0, n \geq 1, k\in\mathbb{Z}$

Hello StackExchange Community, This is my first post on the forum. Please forgive me for any errors with formatting and my expressions. I am working on the following proof: Show $$x^n \geq ...
2
votes
3answers
49 views

Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$

Problem: For positive integers $n,j,k$, prove that the following holds: $$\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$$ I simply ...
1
vote
3answers
34 views

How to prove that the cross product of a countable and uncountable set is uncountable?

so my question is, how can you prove that ${\Bbb Z}$ x ${\Bbb R}$ is uncountable? So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm ...
0
votes
2answers
37 views

How to prove this product rule?

If $f,g:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ are differentiable in $x_0\in\Omega$ ($\Omega$ is open), then the function $(f*g)$ is differentiable in $x_0$ and: $(f\cdot ...
0
votes
0answers
47 views

derivative $1 \over x$ -proof

proving $\frac{1}{x}$ by definition $$(\frac{1}{x})'=lim _{h \to 0} {\frac{1}{x+h}-\frac{1}{x}\over h}=lim _{h \to 0} {\frac{x-x-h}{(x+h)x}\over h}=lim _{h \to 0} {\frac{-h}{(x+h)x}\over h}=lim ...
1
vote
1answer
40 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
1
vote
1answer
53 views

Methods to prove that a function is continuous

Although I seem to understand the concept of continuity in connection with functions, I am often stuck proving that particular functions are continuous. I think the epsilon-delta definition is the ...
1
vote
1answer
25 views

Showing there exists a complex differentiable function $g$ satisfying $g(z_0)=z_0$, with $g'(z_0) \neq 0$ and that $h(g(z))=(z−z_0)^{−m}$.

This is a follow up to a previous question: (Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.) I'm trying to ...
0
votes
1answer
33 views

Prove that if function f is monotonic, then it one-to-one

What I have so far: Suppose $f$ is monotonic. It is therefore either increasing or decreasing. Proof for increasing: If $f$ is increasing, then $f(x_1) <f(x_2)$ whenever $x_1 < x_2$, which ...
1
vote
0answers
33 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...