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.

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Epsilon-delta limit proof, but delta turns out to be negative

Have to prove that $$\lim_{x \to 0} \sqrt{4-x} = 2$$ Using epsilon-delta definition. So have to show that $\forall\varepsilon>0, \exists\delta>0$ such that $0<|x|<\delta \Rightarrow |\sqrt{...
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Exercise 4, Section 32 of Munkres’ Topology

Show that every regular Lindelöf space is normal. My attempt: Let $A$,$B$ be closed in $X$ such that $A \cap B=\emptyset$. By chapter 7 theorem 2.2 of Dugundji topology (equivalent definition of $T_3$...
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Help with Liouville's formula (putting it into "easier words")

I am trying to break down Liouville's formula into "my own words" to see if I am fully understanding it. Does the below steps look ok, especially the variable replacements? I seem to have a ...
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Given group with operation $a ⊗ b = a + b − a · b$ Which of the group axioms are satisfied. [duplicate]

Given group $\mathbb{Z}\backslash\{1\}$ with operation $a \otimes b = a + b − a · b$. Here $+, ·$ denote the normal operations of addition and multiplication in $\mathbb{Z}$. Which of the group axioms ...
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Differential $1$-form and proof of an open disc and open circular annulus not being diffeomorphic

There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ ...
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How to prove that $\forall a \forall b \exists x(a+x=b \lor b+x=a)$

The universe is the set of natural numbers including 0, defined by the Peano Axioms. I tried and failed to prove this by induction on b.
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How do I prove that $\forall a \forall b(\neg a<b\iff b\leq a)$ [duplicate]

The universe is the set of natural numbers including 0, which are defined in accordance with the Peano Axioms. We define the inequalities as: $a\leq b \iff \exists x(a+x=b)$ $a<b \iff \exists x(a+x=...
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Proof verification of exercise 7(a), section 31 of Munkres’ topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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What are some strategies to write a proof that can be easily comprehended?

There are already a lot of articles on how to correctly write a proof. In contrast, assuming that I already have a very hard proof written, I am interested in rewriting the proof to make it easy to ...
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How to proove $\lim_{x \to a}[f(x)g(x)] = 0$ if $\lim_{x \to a}g(x) = 0 $

I am working with "Calculus with Analytic Geometry" from the author Leithold and I came to the execrcise where I have to proove that $\lim_{x \to a}[f(x)g(x)] = 0$ for $\lim_{x \to a}f(x) = ...
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How to prove the following statement regarding the successor function and addition of natural numbers?

Natural numbers (including 0) and the successor function are defined as per the Peano Axioms (you can check them on wikipedia). Addition is defined recursively as follows: $a+0=a$ $a+S(b)=S(a)+b$ With ...
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Comprehending the proof for the addition of limits

The theorem for the addition of limits: $\lim_{x \to a}[f(x)\ \pm\ g(x) ] = \lim_{x \to a}f(x)\ \pm\ \lim_{x \to a}g(x) = K + L$ is often proven using the the idea of getting the module of the ...
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Help me rewrite Theorem 3 from Serge Lang's Basic Mathematics using Velleman's 'given–goal diagrams'

I want to apply the proof strategies and overall scratch work diagram based framework introduced in the book "How to Prove It" in order to rewrite the following theorem: Theorem 3. Any ...
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Proof check: Version of IVT from Spivak's Calculus

Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, then there is some number $x \in [a.b]$ such that $f(x) = 0$. I have rewritten Spivak's proof using my own understanding of ...
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advice about how to write proofs and demonstrations in real analysis [closed]

3 months ago i started in real analysis, then i detected my gapp in logic and i read a book named "a mathematical introduction to logic" by Herbert. Then i started feeling like i'm drowing ...
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Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
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Give an equation for a line that contains no lattice points.

Give an equation for a line that contains no lattice points. Explain how you know it contains no lattice points. This problem is taken from section 5.1 exercise Q.9. of book: Number theory, by: James ...
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If a function is positive, prove that its limit is also positive [closed]

To solve this, I though I should use the definitions of left and right limits, not sure how that would work though. I really have no idea where to start on this. Assume $g : (\alpha, \beta) \to \Bbb R$...
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summer reading suggestions

I have just finished 2nd year of my maths degree and I was wondering if anyone had any good recommendations for summer reading, I want to make sure I keep practising so that I don't feel underprepared ...
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Prove that every cycle graph $C_n$ has $n$ edges

I need to prove this directly and by induction. I do not even know where to start. Question: A cycle graph $C_n$ is a connected graph with $n$ vertices, such that each vertex is adjacent to exactly ...
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2 votes
2 answers
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Prove the number of elements of order $n$ in $\mathbb{Z}_n$ is $\phi (n)$

Prove the number of elements of order $n$ in $\mathbb{Z}_n$ is $\phi (n)$, where $\phi (n)$ is the Euler Totient Function. The hint for this problem says "You need to decide which $[a] \in \...
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Exercise 7(c), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (c) Show that if $X$ is ...
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If I define addition in the following way, how can I prove that it's commutative?

$a+b=a$, if $b=0$ $a+b=S(a)+S^{-1}(b)$, if $b\not=0$ Here a and b are natural numbers defined according to the Peano axioms, while S represents the successor function. Basically, I am trying to prove ...
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Exercise 7(b), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (b) Show that if $X$ is ...
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2 votes
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Exercise 7(a), Section 31 of Munkres’ Topology

Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is compact for each $y \in Y$. (Such a map is called a perfect map.) (a) Show that if $X$ is ...
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2 votes
2 answers
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How to prove the product of first n consecutive odd numbers is a square less than another square?

I have observed for first few values of consecutive odd numbers, the result is always of the form: $m^2 - n^2$, where $m$ and $n$ are two distinct positive integers. That is: $$1\cdot 3\cdot 5 \cdot 7\...
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Linear mapping rank question

Let $X:U \rightarrow V$ be linear, where $U, V$ are vector spaces. And $Y: T \rightarrow T$ be linear, and $\ker(Y)=\{\vec{0}_T\}$. Prove $rank(Y \circ X)=rank(X)$. I started by doing an example. Let $...
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How do formal proofs work and relate to interpretations?

As far as I know statements in formal logic are written in a (formal) alphabet which are just symbols, where the allowed sentences have to follow certain rules. If they do, they are called well formed....
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Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.

I have proved in what follows that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$ and I would appreciate if someone would check my proof and/or point out how to ...
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Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

Proof by contrapositive: Let $x \in \mathbb{Z}$. Assume that $2 \nmid x$. Thus, $\forall k \in \mathbb{Z}$, $2k \neq x \Rightarrow (2k)^3 \neq x^3 \Rightarrow 8k^3 \neq x^3 \Rightarrow 2(4k^3) \neq x^...
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2 votes
1 answer
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How to prove $\forall x\; (\phi (x)\land \psi (x) ) \rightarrow \forall x\; \psi (x)$ without using the completeness theorem?

The statement $\forall x\; (\phi (x)\land \psi (x) ) \rightarrow \forall x\; \psi (x)$ is valid, that is it is true in any structure. Hence, for any $\sum \subseteq Form_{\mathcal{L}}\; \sum \models \...
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Exercise 6, Section 31 of Munkres’ Topology

Let $p \colon X \to Y$ be a closed continuous surjective map. Show that if $X$ is normal, then so is $Y$. [Hint: If $U$ is an open set containing $p^{-1}(\{ y \} )$, show there is a neighborhood $W$ ...
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proof that norm two of a matrix to the power of two is less than or equal to norm one multiplied by norm infinite [closed]

I want to proof that enter image description here I think im supposed to multiply both sides of the equation by x and than try to expand them(sorry if my english isn't the best im from Iran). but the ...
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Proof for weighted sample central moments

I think the question is valuable for the community as I could not find out any close related topic on stackexchange or even internet. We have a sample of observations and we decided to replicate some ...
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Exercise 12, Section 26 of Munkres’ Topology

Let $p:X\to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y\in Y$. (Such a map is called a perfect map) Show if $Y$ is compact, then $X$ is compact. [Hint: ...
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Missing step in proof : if prime number $p\mid ab$, then at least one of $a, b$ is divisible by $p$. [duplicate]

Request the needed missing step proof below. Given $a, b\in \mathbb{N}, p\mid ab$, given a prime number $p$, then at least one of $a, b$ is divisible by $p$. Proof: By hypothesis, $ab = cp$. Wlog, $p\...
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1 vote
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Show that the set of $n^{th}$ roots of unity form a subgroup under the group $S^{1}$

So, the set of $n^{th}$ roots of unity are defined as: $C_{n} = \{e^\frac{2 \pi k i}{n} \mid k \in \{0,...,(n-1)\}\}$ And the group $S^1$ is defined to be: $S^1 = \{z \in \mathbb{C} \mid \lvert z \...
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3 answers
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Questions about Proof by Induction [closed]

To do a proof by induction, it is necessary to prove (1) the proposition $P(n)$ is true for $n=1$ (2) if the proposition $P(n)$ is true for $n=k$, then $P(n)$ is true for $n=k+1$ My questions are (1) ...
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Exercise 1, Section 32 of Munkres’ Topology

Show that a closed subspace of a normal space is normal. My attempt: Approach(1): Let $\{y\}$ be a singleton set in $Y$. Since $X$ is $T_1$, $\{y\}$ is closed in $X$. $\{y\} =Y\cap \{y\}$. By theorem ...
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1 vote
1 answer
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Let $\mathcal{A}= \{A_n = \left(0, \frac{n}{n+1} \right) \ | \ n \in \mathbb{N} \}$. Prove that $\bigcup \mathcal{A}\subseteq (0,1)$.

Let $\mathcal{A}= \{A_n = \left(0, \frac{n}{n+1} \right) \ | \ n \in \mathbb{N} \}$. Prove that $\bigcup \mathcal{A}\subseteq (0,1)$. Can you verify this solution? Let $x \in \bigcup \mathcal{A}$. ...
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3 answers
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Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$? My attempt: It seems to me that $(0,1) \subseteq \left(0, \frac{n}{n+1}\right)$ is a valid argument for some $n \in \...
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2 votes
1 answer
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Proving the function $f(x) = x^2 + ax + b$ is not injective. Does my proof make sense?

The question I’ve been working on is: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be the function defined by $f(x)=x^2+ax+b$, where $a,b\in\mathbb{R}$. Prove that $f$ is not injective. Here's what I've ...
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1 answer
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Let $\Gamma \neq \emptyset$. Also $\Gamma \subseteq \Omega$. Show that $\bigcap \Omega \subseteq \bigcap \Gamma.$

Let $\Gamma$, $\Omega$ be collection of sets and $\Gamma \neq \emptyset$. Also $\Gamma \subseteq \Omega$. Show that $\bigcap \Omega \subseteq \bigcap \Gamma.$ My solution: Let the set $X \in \Gamma$ ...
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1 vote
1 answer
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How to write maths

In the sentence below, which expression is better to choose: "their multiplications" or "their multiplication" : "This identification allows us, first, to present a new polar ...
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1 vote
0 answers
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Understanding Leray-Hirsch theorem from Bott and Tu.

What does the statement actually mean? Could anyone please share some ideas on it? Also how does it follow from Künneth formula? Any help in this regard would be warmly appreciated. Thanks for your ...
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Prove that the following recursive functions (defining addition) are equivalent for all natural numbers

$S$ represents the successor function. $S(0)=1, S(1)=2, S^{-1}(3)=2$ and so on. First definition: $$a+b=sum1(a,b)\\ sum1(a,b)=a, \rm{if\ } b=0\\ sum1(a,b)=S(sum1(a,S^{-1}(b))), \rm{if } \neg b=0$$ ...
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0 answers
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For which values of $n$ does there exist $n$ real $n\times n$ matrices $A_1,\cdots,A_n$ such that $A_1v,\cdots,A_nv$ are always linearly independent? [duplicate]

Here's a question that I have been stuck trying to figure out the right answer AND write a vigorous proof for weeks. Here's the question: For which values of $n$ does there exists $n$ real $n\times n$ ...
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2 answers
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How do I prove formally that for all natural numbers $a\cdot S(c)=b\cdot S(c)\implies a\cdot c=b\cdot c$

Natural numbers, addition, multiplication, and the successor function S, are defined in the wikipedia article regarding Peano axioms. https://en.m.wikipedia.org/wiki/Peano_axioms Originally I was ...
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2 answers
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For all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational.

How can I show that for all natural numbers $n$, if $n$ is odd, then $\sqrt{15^n}$ is irrational? I have tried to use a proof by contradiction to no avail. I have gotten decently far with a proof by ...
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What is the negation of the Universal Conditional Statement

This being the UCS: (∀x ∈ D)(P(x) → Q(x)) How would I negate the statement? And how would I push this negation inside the formula as far as it would go?
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