# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### Is my proof that $(A^n)^{-1} = (A^{-1})^n$ correct?

I am still learning Linear Algebra at it's basic levels, and I encountered a theorem about invertible matrices that stated that: If $A$ is an invertible matrix, then for $n=0,1,2,3,..$. $A^n$ is ...
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### Prove that there is no number that divides both n and n+1

Statement There is no number $x > 1$ that divides both $n$ and $n+1$. Proof (my attempt) Indirect proof: \begin{align} x\mathbin{\vert} n & \implies n = xt_1 \\ x\mathbin{\vert}(n+1) ...
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### Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but …

Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but $$\mu\left(\bigcap_{n=1}^\infty E_n\right)=0$$ Claim: Let \...
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### A continuous function $f$ from a closed bounded interval $[a, b]$ into $\mathbb{R}$ is uniformly continuous

I am reading (as a supplement) the book Basic Real Analysis, by Anthony Knapp. Before I proceed into reading a proof, I want to be sure that the result seems obvious. Yet, I am having trouble seeing ...
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### Show that $\gcd(a,b)=\operatorname{lcm}(a,b)$ if and only if $a=b$.

I know how to prove $a=b$ only if $\gcd(a,b)=\operatorname{lcm}(a,b)$, but I don't know how to prove the "if part". Can anyone help me?
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### Proving the so-called “Well Ordering Principle”

Is there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
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### 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 ...
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### Prove that such an inverse is unique

Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$ $z^{-1}$ is also often written as $1/z$
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### Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
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### Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
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### prove: if n is even, then n+1 is not even

This proof seems so simple that it's hard (if that makes any sense.) based on the definition, n is even iff there exists k such that n = 2k. What I really want to say is (big picture) By definition,...
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### Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
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### Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly increasing ...
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### how to prove $\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$

i am asked to prove this statement: $$\left(\frac{n}{3}\right)^n\leq\frac{1}{3}n!$$ Now after several attempts, i am lost not knowing where and how to start. if I use induction, i am stuck on ...
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### Proof of $\gcd(a,b)=ax+by$

Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs? $a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be ...
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### How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
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### 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 ...
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### How to prove $4(n!)>2^{n+2}$ for $n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
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### Prove the inequality $n! \geq 2^n$ by induction

I'm having difficulty solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this by induction. I started like this: The lowest natural number where the ...
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### Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$

I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with $$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$ First thing I want to ...
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### Can you prove why Popsicle Stick Multiplication works?

This is a unique way of multiplying numbers by using sticks. Let's call it "Popsicle Stick Multiplication". Or maybe "Linear Algebra" quite literally. Take a look at both images that I've drawn ...
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### prove $\lceil{x}\rceil=-\lfloor-x\rfloor$

i am trying to prove that $\lceil{x}\rceil=-\lfloor-x\rfloor$, but having difficulties to prove. the definitions are: $\lceil{x}\rceil:=m-1<x\leq m$ and $\lfloor{x}\rfloor:=n\leq x<n+1$. how ...
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### How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
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### 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 $B\subseteq A$. (...
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### Supposed proof of dirichlets theorem on primes

I think theirs somthing wrong with this proof as it was not hard to create, if someone could find a mistake I would greatly appreiciate it: Define a function $[k\equiv b \bmod a]$, to be equal to ...
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### help me understand a line in an “$A^TA$ is positive, semi-definite” proof

I am looking at a proof for why $A^TA$ is positive semi-definite when $A$ is $n\times n$ and it has this line. $$v^TAA^Tv = A^Tv \cdot A^Tv ≥ 0.$$ I understand what $v^TAA^Tv$ means and the purpose ...
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### prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective

Question would be: prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective. after thinking, I came to the conclusion that it's a proof. tried to prove it but it looks not ...
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### Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
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### Show that if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$

Prove if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$. Obviously, $x,y \in \mathbb R$. I know this is rather simple. It is more about the process than this example. Is it logically correct to do ...
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### How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
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### Sipser Pumping Lemma Clarification

In a Theory of Computation book I am using, the explanation of Pumping Lemma is not bad, but some parts of it are not clear to me. Here is the Definition of Pumping Lemma: If A is a regular language,...
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### What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
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### Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
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### The author of my book simplifies his solutions to an extent that I am uncomfortable with, so are my solutions to homework over doing it?

This question can be summarized as: How explicit does one need to be when writing proofs? To what extent can one implicity write a proof safely? The first chapter of our text in elementary discrete ...
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### natural language proof assistant

I was wondering whether there has been any attempt to create a proof assistant that you write in it, in english, I mean you write your proof the usual way in TeX(maybe use a 'simpler english') then ...
### Prove A is an open set if and only if $A \cap Bd(A) = \emptyset$
Prove A is an open set if and only if $A \cap Bd(A) = \emptyset$ Here is my start: Suppose A is an open set. We know $X-A$ is closed. Need to show $A \cap Bd(A) = \emptyset$ Let $x \in A$. ...
For British maths style, is this punctuation OK? so if $x=-3$, then $\left|x\right|=3$, and if $x=7$, then $\left|x\right|=7$, etc with commas before "then" and "and".