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

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1answer
74 views

Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
-2
votes
2answers
38 views

Prove by induction that $\sum_{i=1}^n i \geq \frac{n^2}{2}$ [closed]

Can someone show me a formal proof of this exercise ? \begin{equation} \sum\limits_{i=1}^n i \geq \frac{n^2}{2}, \quad \forall n \in \mathbb{N}. \end{equation} Thanks to anyone who can help! :)
0
votes
1answer
20 views

Prove a relation of a distance function

I had to do an exercise with this function: $$ d_M:\Bbb C \rightarrow \Bbb R, \quad z \rightarrow inf\{ |z-w|; w \in M| $$ with $\emptyset \neq M\subset \Bbb C$. First I proved that this function is ...
0
votes
1answer
116 views

Proof that a certain language is Turing Decidable

$$L_1 = \{\langle R,S \rangle \mid \text{$R$ and $S$ are regular expressions and }L(R) \subseteq L(S)\}$$ $$L_2 = \{\langle M\rangle\mid \text{$M$ is a DFA that accepts $w^r$ whenever it accepts $w$} ...
1
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2answers
93 views

Epsilon-Delta proof for continuity

I have a lot of trouble figuring out how to work with this proof technique for continuity. The definition says: $$ \forall \varepsilon \space \exists \delta \quad |x-a|\lt \delta \quad \Rightarrow \...
1
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1answer
39 views

Looking to receive feedback on elementary proofs in topology

I'm looking to receive some feedback on a couple of proofs I wrote verifying the discrete and trivial topologies and another simple topology. I'm inexperienced with proof (in the sense that I haven't ...
1
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1answer
23 views

Proving path length, transitive closure

Set A is finite with n elements. Suppose a and b are elements of a set A with a != b. Let R be a relation on the set A so that there is a path from a to b of length at least 1. Show there is a path ...
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1answer
38 views

Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
2
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2answers
29 views

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field?

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field ? I am claim that it is not. My reasoning is this... What I am ...
0
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1answer
24 views

Is this a valid proof of set membership

Let $S=\{x \in\mathbb{Z}: x\geq0, x=b-a ×m$ for some $m\in\mathbb{Z}\}$. Prove that if $b\geq0$ then $b$ is an element of $S$. Pf: suppose $b\geq 0$ Let $a$ be an integer define $b=b-a×m$ Where ...
2
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2answers
75 views

Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
2
votes
2answers
31 views

On the limit of $f(n)$, specifically having to do with integration of an iterated $\arctan$

Assume we are given that $A_n(x)$ denotes $n$ iterations of $\arctan(x)$, for example $A_2(x)=\arctan (\arctan(x))$ If $$f(n)=\int_{0}^n A_n(x)\space \text{d}x$$ I am looking for a rigorous proof ...
2
votes
2answers
59 views

Is the writing of the proof ok?

Problem. Let $f:(0,\infty)\to\mathbb{R}$. Prove that, $$\displaystyle\lim_{x\to\infty}f(x)=L\iff\displaystyle\lim_{x\to0 +}f\left(\dfrac{1}{x}\right)=L$$ My Solution. Let us assume that $\...
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1answer
63 views

Generalized DeMorgan's Law proof

We wish to verify the generalized law of DeMorgan $(\bigcup_{i \in \mathcal{I}} A_i)^c = \bigcap_{i \in \mathcal{I}} A_i^c$. Let $ x \in (\bigcup_{i \in \mathcal{I}} A_i)^c$. Then $x \notin \...
0
votes
1answer
37 views

Linear Algebra Proof - Columns of Matrix Linearly Independent & Determinant

How can I prove that if the columns of matrix A are linearly independent, then det(A) does NOT equal zero? This is a question on my exam review and I have no idea how to go about proving this. Any ...
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votes
1answer
49 views

Decide whether logical formula is a tautology [duplicate]

How do we decide whether the formula in predicate logic is a tautology? Is there some universal way to decide and prove it? Let's have an example: $$ \forall x \forall z \exists y\,(P(x,y) \lor P(y,...
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2answers
105 views

How to check, whether the formula is a tautology

How do we decide whether the formula in predicate logic is a tautology? Is there some universal way to decide? Let's have an example: ...
1
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1answer
60 views

Proof: Fibonacci Sequence (2 parts)

Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers. I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely ...
0
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0answers
4 views

Approximating semicontinuous functions by continuous functions. [duplicate]

Let $f=f(x):[0,1]\to\mathbb{R}$ be a upper (or lower) semicontinuous function, i.e., $$\limsup_{j\to\infty}f(x_{j})\le f(x)\quad\text{for $x_{j}\stackrel{j\to\infty}{\longrightarrow}x$}$$ (or $\...
3
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0answers
40 views

Leibniz Notation for the Derivative of a Function

I am writing a professionally-written proof, and I have come across a bit of an issue regarding how to write the derivative of a function $H(t)$ with respect to $t$. Is $\frac{dH}{dt}$ an acceptable ...
2
votes
3answers
61 views

Proving $\sqrt{2}x-\sqrt{x^{2}+1} \geq \frac{\sqrt{2}}{2}\ln{(x)}$

How can I prove that $$ x\sqrt{2}-\sqrt{x^{2}+1} \geq \frac{\sqrt{2}}{2}\ln{(x)} $$ It's a derivation-based process if I remember correctly, however I was unable to prove it correctly.
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2answers
48 views

Prove that a sequence is bounded/unbounded

I'm trying to do a maths problem which requires me to determine whether a sequence is bounded or unbounded and then it wants me to prove my answe. I know that it's bounded but I've no idea how to ...
1
vote
1answer
24 views

How to define two functions in a clear and standard way?

I am working on a question and before I ask it, I wanted to get help in defining two functions clearly in a standard way. Here are the two functions: $f(a,x,p)$: count of the number of pairs $k,k+2$...
3
votes
7answers
216 views

Proving $\cos(x)^2+\sin(x)^2=1$

I need to prove that $\cos(x)^2+\sin(x)^2=1$ Here's how I started (using the Cauchy product): \begin{align} \cos(x)^2+\sin(x)^2 &=\sum_{k=0}^{\infty}\sum_{l=0}^k(-1)^l\frac{x^{2l}}{(2l)!}(-1)^{k-...
3
votes
1answer
43 views

$3x + 1$ is even iff $5x-2$ is odd

I'm asked to prove 'Let $x \in Z$. $3x + 1$ is even iff $5x-2$ is odd'. I have the following proof techniques in my toolbox: trivial/vacuous proofs (not so relevant in this case), direct proof and ...
0
votes
1answer
68 views

Any composite natural number divides the product of two smaller natural numbers

Let $\alpha$ be a composite natural number not equal to 4. Show that $\exists m,n \in \mathbb{N}$ such that $ 1 < m < n < \alpha$ and $\alpha|mn$. This is my proof so far. Split it up into ...
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votes
2answers
58 views

greatest common divisor proof [closed]

Suppose $\gcd(a,b)=1$. Does this necessarily imply that $\gcd(a,2b)=1$? If $\gcd(a,b)=1$ then $\exists x,y \in \Bbb{Z} (ax+by =1)$ I don't see how to manipulate this equation to give me $ax+2by \...
1
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1answer
34 views

Proving relatively prime using GCD definition

I'm stuck on proving a few facts. My thoughts for each question are below, what do you guys think? Let a and b be positive integers. Prove that $2^a$ and $2^b-1$ are relatively prime by showing that ...
1
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1answer
138 views

Prove that $x^3-x = y^2+1$ has no integer solution

Prove that $x^3-x = y^2+1$ has no integer solution: I began the proof by case distinction considering the cases if x,y are both even, if x,y both odd, if x even, y odd and the last one if x odd and y ...
0
votes
0answers
25 views

Proof on exponential Relation R

Prove that $(R^a)^b = R^{ab}$ for any integers $a,b >= 1$. A handy fact: The connectivity relation $R^*$ consists of the pairs $(x, y)$ such that there is a path of length at least $1$ from $x$ to ...
0
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0answers
35 views

A set A is infinite if and only if there is a bijection from the set A to a proper subset of A. [duplicate]

I'm just starting my journey into proof writing and I don't really know how to do this. More specifically I think I want a proof of the fact that every infinite set A is Dedekind-infinite (i.e. that ...
0
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2answers
37 views

Implication vs Equivalence in proofs

I understand the definition of both the implication and equivalence signs. When I get asked to prove something, I will probably have to use both implication and equivalence logic. My question is if it'...
1
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4answers
142 views

Proof of greatest common divisor [duplicate]

The greatest common divisor of two positive integers $a$ and $b$ is the largest positive integer that divides both $a$ and $b$ (written $\gcd(a, b)$). For example, $\gcd(4, 6) = 2$ and $\gcd(5, 6) = 1....
1
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1answer
35 views

Simple examples of applications of converse, contrapositive and inverse used in mathematical proofs rather than logic.

While learning simple logic in high school, I remember learning about converse, contrapositive and inverse (maybe some others as well). Yet, I don't seem to recall their usage for proofs (only ...
1
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1answer
40 views

Proof via mean value theorem

Suppose $f$ is differentiable with $f(0)=0$; $f'(x)<0$ for all $x<0$ and $f'(x)>0$ for all $x>0$. Prove $f(x)\geq 0$ for all $x\in\mathbb{R}$. It's pretty clear to me that $f$ decreases ...
2
votes
1answer
34 views

How to prove ~ is an equivalence relation

In a metric space $M$, declare $x \sim y$ to mean that there is a continuous function $\gamma : [0, 1] \rightarrow M$ such that $\gamma(0) = x$ and $\gamma(1) = y$. Prove that $\sim$ is an equivalence ...
1
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1answer
28 views

Proof: An accumulation point if each neighborhood contains point not itself

I want to confirm my proof of this lemma. Lemma: Let $S$ be a set of real numbers. Then $a$ is an accumulation point of $S$ if and only if there each neighborhood of $a$ contains a member other than ...
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3answers
53 views

$xf'(x) = αf(x)$. How to prove that $f(x) = cx^\alpha$?

Let $f$ be a differentiable function such that $xf'(x) = \alpha f(x)$ for all $x > 0$. How do I show that $f(x) = cx^\alpha$ for some constant $c$? I have $f'(x) = \alpha f(x)/x$ , and I can see ...
2
votes
2answers
92 views

Proving $n\sin(\frac{\pi}{n})<\pi<n\tan(\frac{\pi}{n})$ ; obtaining results from it.

I was reading The Simpsons and the Mathematical Secrets when I encountered the story of $\pi$. It mentions how Archimedes devised a method to place a lower and upper bound on $\pi$ by bounding a ...
0
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1answer
94 views

Question about this specific proof of $\forall a\in G$ $aH=Ha$ implies $H$ is normal

I've just tried to come up with a proof of the above statement but I feel like something is not quite right. The question I have isn't about proving the statement, I've found that in lots of places, ...
0
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2answers
20 views

Simplest Proof for an Elementary Number Theory Condition

I have to proove in the most 'primitive' way the following: if $ab=ac$ then either $a=0$ or $b=c$. I could think only about the following solution: considering the given $ab=ac$, let's subtract from ...
1
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1answer
67 views

Let A be a denumerable set. Prove that the set $\{B:B\subset A\}$ and cardinality of B=1 of all 1-element subsets of A is denumerable.

So my original idea was to show that the countable union of countable sets is countable since I know that each set has one element. I'm not exactly sure how to start this off though. Thanks in advance....
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2answers
59 views

Prove or disprove: If $f(x): \mathbb{R} \rightarrow \mathbb{R}$ is bounded above, then $f(x+10)$ is also bounded above.

We want to prove or disprove: If $f(x): \mathbb{R} \rightarrow \mathbb{R}$ is bounded above, then $f(x+10)$ is also bounded above. 1) I need to first identify if this statement is true or not true. ...
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1answer
77 views

If $\{B_{i} : i\in N\}$ is a denumerable family of pairwise disjoint distinct finite sets, then $\bigcup_{i\in N} B_{i}$ is denumerable.

So for this proof I'm thinking of creating a bijection from N to $\bigcup_{i\in N} B_{i}$ , but I'm not sure how I should go about doing this. I thought about listing out a couple of sets from $\{B_{...
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1answer
132 views

Graph theory: proving that a graph with specific property is bipartite

I have been given the following problem on an exercise sheet: Let $G$ be a graph with $n$ vertices with the property that for each $k ≤ n$, every set of $k$ vertices contains a subset of size at ...
1
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2answers
45 views

Common methods for proving: Existence proof

For every real number x with $x\neq -1$ there exists a real number y such that $ \frac{y}{y+1}=x $. $ \forall x\in \Bbb R\setminus (1) \ \ \exists y \in \Bbb R : \frac{y}{y+1}=x \\, x \neq -1 $ $ ...
1
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1answer
33 views

Proof strategy about a property of triangular matrices

Is it by mathematical induction the best way to prove that the determinant of an upper (lower) triangular matrix is the product of the elements of the main diagonal? Actually, I am wondering about ...
0
votes
1answer
43 views

Let $f(x) = x^2 - 4x - 12$ prove that $f$ is not uniformly continuous on the set $[-2,∞)$

so I know that to show this is NOT uniformly continuous then I need to show that $\exists \epsilon>0$ $\forall \delta>0 \exists x,y\in [-2,\infty)$ such that $(|x-y|<\delta ~~\&~~ |f(x) - ...
1
vote
1answer
46 views

Proving the area of a triangle within a triangle

Consider a triangle with vertices ABC, we pick a point C' on the line segment AB in such a way that |BC'|=2|AC'|. Similarly, we pick a point B' on the line segment AC and a point A' on the line ...
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votes
2answers
121 views

Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular, without using $\lambda_1\cdot\ldots\cdot\lambda_n = det(A)$. [duplicate]

I would like to know if there is any proof without using the fact that: $$\lambda_1\cdot\lambda_2\cdot\ldots\cdot\lambda_{n-1}\cdot\lambda_n = det(A)$$ I managed to prove that if $\lambda = 0$ then, ...