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

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2answers
41 views

Show that well-ordering is not a first-order property.

Problem description: Show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma \cup ...
0
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3answers
40 views

Does proof proceed from left to right?

Very simple question: if we're asked to prove that $a=b$, do we start with $a$ and then find $b$ from $a$? Does going the other way around count as a formal proof? The exercise in question is this: ...
1
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3answers
32 views

How to prove that the g.c.d is equal to the prime factorization raised to the minimum of two powers

for the prime factorization of $a$ and $b$ as $$a = p_1^{\alpha_1}p_2^{\alpha_2}\cdots{p_t}^{\alpha_t}$$ and $$b = p_1^{\beta_1} p_2^{\beta_2} \cdots p_s^{\beta_s}.$$ I want to prove that $d = (a,b)$ ...
1
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1answer
38 views

show that $3^{1974} + 5^{1974} \equiv 0 \bmod 13$

show that $3^{1974} + 5^{1974} \equiv 0 \bmod 13$ My attempt with this question was to use Fermate Little's THM. But I do not understand how to properly use it for this question. Can some one show me ...
0
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1answer
18 views

Proof of equivalency in disjoint sets.

Prove, If A, B, C, and D are sets with |A|=|B| and |C|=|D| and if A and C are disjoint and B and D are disjoint, then |A ∪ C|= |B ∪ D|. Would I start this proof using the definition of disjoint ...
2
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3answers
136 views

Is it acceptable to use “But” in a proof that doesn't use contradiction?

I have recently read a lot of proofs that like to say "But..." right before the punchline. I feel that the word "But..." should be used if what follows is contradictory in some way, as in proofs by ...
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1answer
16 views

How to specify a function with flexible domain but same range?

As an example, I could be interested in functions that operate on $\mathbf{R}$ and $\mathbf{R}^2$. One way to say this is "all functions $f:\mathbf{R} \to \{0,1\}$ and all functions $f: \mathbf{R}^2 ...
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2answers
45 views
2
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1answer
58 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
1
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2answers
71 views

Proving some functor is adjoint to another. What to do with naturality condition?

Whenever I want to prove that some functor is (left/right) adjoint to another, I (mostly using hom-set definition) go on smoothly to prove the "isomorphism of the corresponding hom-sets", until it ...
0
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1answer
17 views

Inductive proof structure

To prove a statement about recursive series, is it correct to use an inductive proof structure showing that if $n = k$ and $n = k + 1$ are true then $n = k + 2$ holds true, and then prove the ...
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1answer
14 views

How to prove $|x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$ with the following condition?

How to prove $\forall \epsilon\in \Bbb{R^+},\exists \delta\in\Bbb{R^+},\forall x\in\Bbb{R^+}, \forall y\in\Bbb{R^+}, |x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$. My try: Pick ...
2
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0answers
18 views

How to prove $|n-m|\lt d\implies |f(n)-f(m)|\lt e$ for the following condition?

How to prove $\exists n\in\Bbb{N},\forall e\in \Bbb{R^+},\exists d\in\Bbb{R^+},\forall m\in\Bbb{N}, |n-m|\lt d\implies |f(n)-f(m)|\lt e$ if $f(x)=\lfloor \frac{n}{3}\rfloor$. I don't know how to ...
0
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2answers
31 views

Is it true that $X\iff Y$ is equivalent to $[X\land Y]\lor [\neg X \land \neg Y]$?

Is it true that $X\iff Y$ is equivalent to $[X\land Y]\lor [\neg X \land \neg Y]$? I don't see anything wrong with the statement. Could someone confirm?
0
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0answers
18 views

How to prove the following facts regarding the matrix

Let $X$ be a connected graph on $n$ vertices and $n$ edges. Let $Q$ be its edge incidence matrix.If $T\subset \{1,2 ,n\}$ with $|T|=n-1$ then $\det (Q[1,2 ,n-1\mid T])=^+_- 1$ if and only if the ...
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2answers
52 views

How to generalise this complex equation?

I am trying to generalise the statement for $n$ complex numbers: For any complex numbers $a,b,c$ with property $|a|=|b|=|c|=r\neq 0$. Prove $|\frac{ab+bc+ca}{a+b+c}|=r$ I proved this by showing ...
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2answers
33 views

Proof by Induction: Stuck

I feel as if this should be really really easy but is my brain getting there...? No... So here it is: Prove by induction that if $b$ is an odd number and $n$ is a positive integer, then $b^n$ is ...
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5answers
58 views

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove ...
1
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1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
0
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1answer
38 views

Given sequences $(x_n)$ and $(y_n)$, define $(z_n)$ as $z_{2n-1} = x_n$ and $z_{2n} = y_n$. If $\lim x_n = \lim y_n = a$, so $\lim z_n = a$

Given sequences $(x_n)$ and $(y_n)$, define $(z_n)$ as $z_{2n-1} = x_n$ and $z_{2n} = y_n$. If $\lim x_n = \lim y_n = a$, so $\lim z_n = a$. I would like to know if my attempt and writing is ...
0
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2answers
31 views

Proof of non-uniform convergence of $x^n$

The wikipedia page on uniform convergence indicates that $f_{n}:[0,1]\to[0,1]$ with $f_{n}(x):=x^{n}$ converges pointwise but not uniformly to $$ f(x)= \begin{cases} 0,\quad x \in [0,1)\\ 1,\quad ...
0
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1answer
34 views

Least involved proof for continuos functions => uniform continuos functions on [a,b]

I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is: If f is continuous on a closed interval [a,b], then f is uniformly ...
1
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2answers
31 views

Proving a Bunch of Statements

Been doing practices problems so... Consider the following statements: (a) Define what it means for a real number to be rational and for a real number to be irrational. Answer: I assume that this ...
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2answers
40 views

Proof Involving Irrational and Rational Numbers

Prove that if $xy$ is irrational then at least one of $x$ and $y$ is irrational. Here's what I did: Let $$r=xy$$ Assume $x\in\mathbb{Q}$ and $y\in\mathbb{I}$. Assume $r\in\mathbb{Q}$. Then ...
0
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1answer
44 views

Proof Fragment and Questions

Consider the following proof fragment. There exists an integer k such that $n=3k+1$. Then $n^2=(3k+1)^2=9k^2+6k+1=3(3k^2+2k)+1$ For each of the statements (a), (b), (c) below, answer the following. ...
1
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1answer
28 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
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0answers
51 views

Mathematical proof using sequences

Let $\{x_n\}$ be a bounded sequence. a) Prove that there exists an $s$ such that for any $r > s$ there exists an $M ∈ \Bbb{N}$ such that for all $n ≥ M$ we have $x_n < r$. b) If s is a number as ...
2
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1answer
19 views

All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
0
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0answers
33 views

Mathematical Proof of sequences. Any help is appreciated

Let $\{x_n\}$ be a bounded sequence. a) Prove that there exists an s such that for any r > s there exists an M ∈ N such that for all n ≥ M we have $x_n$ < r. b) If s is a number as in a), then ...
0
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2answers
37 views

How to prove that $\limsup X_n \leq \sup\{X_n\}$?

How to prove that $\limsup X_n \leq \sup\{X_n\}$? I need to prove this and I don't know how to go about doing this. Thank you for any help you can provide.
1
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1answer
28 views

Induction proof (bitstring length)

Theorem : The number of bitstrings with the length $x$ that begin with $1$ and/or end with $0$ is $3 \times 2^{x-2}$. I know there are easier ways to prove this but I must figure out how to do it ...
2
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1answer
47 views

Induction Proof: If $B \subseteq A$, then $|B| \leq |A|$.

Prove by induction that if $A$ is a finite set and $B$ is a subset of $A$, then $|B|≤ |A|$. I can prove the base case with $n=0$ easily, but am stuck as to how to proceed from there.
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0answers
27 views

Theoretical Math Sequence Proof

Suppose that {xn} is a sequences such that every subsequence {xni} has a subsequence {xnmi} that converges to x. Show that {xn} is bounded. I tried to do a proof by contradiction but am not sure if ...
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6answers
33 views

Discrete math induction proof (divisibilty) [duplicate]

How to show that $10^n -(-1)^n$ is always divisible by $11$ through proof of induction?
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1answer
19 views

Infimum and supremum of finite ordered subsets

I am currently taking an introductory proofs course, and I have come across this problem. It's asking to prove the following: Let $S$ be an ordered set. Let $A$ be a non-empty finite subset. Then ...
3
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0answers
31 views

Recreational math dealing with twin primes

This is kinda recreational math with a goal in mind of progressing further toward a proof of the twin prime conjecture. Consider this: We start with a random prime: $109$ $3*109=327$ $327 ...
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2answers
27 views

Prove {$x|P(x) \land S(x)$}$ \cup $ {$x|P(x) \land \neg S(x)$} $= ${$x|P(x)$}

So far, I have only expanded the left hand side to [ {$x|P(x)$}$\cap ${$x|S(x)$} ] $\cup $ [ {$x|P(x)$ }$\cap ${$x|\neg S(x)$} ] and I'm not sure what to do next.
0
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2answers
25 views

Show $∀n≥3$, $2n^2+1 ≥ 5n$

I was able to prove the base case statement, where if you plug in $3$ for $n$ you get: $19 ≥ 15$. Next I supposed an arbitrary value $k$ where $k ≥ 3$ and $2k^2+1 ≥ 5k$. I know that next I need to ...
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0answers
33 views

How to prove this for number theory?

Let $n=p_1^{e1}p_2^{e2}\cdot \cdot \cdot p_k^{ek}.$ Then, $\phi(n)$= $n(1-$ $\frac{1}{p_1})$ (1- $\frac{1}{p_2})$$\cdot \cdot \cdot$ $(1- $ $\frac{1}{p_k})$ Hint: use the following If n has a prime ...
0
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1answer
17 views

Is this a sufficient proof for x-ε<a≤x?

The problem is: If y=Sup(S), show that, for each ε>0, there is a ∈ S such that x-ε < a ≤ x. Proof: Suppose x=Sup(S). Let S={n: n is a real number and n≤x}. Consider x-ε, where ε>0. Since x-ε is ...
0
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4answers
75 views

How to prove that there are no positive integer solutions $(x, y)$ to the equation $x^2 - y^2 = 1$

Prove the following: Theorem. There are no positive integer solutions $(x, y)$ to the equation $x^2-y^2=1$.
2
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2answers
149 views

Prove that additive order is preserved by isomorphisms

I want to show that $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ are not isomorphic by using the fact that $\mathbb{Z}_4$ has one element of additive order 4 (the largest additive order), ...
0
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2answers
81 views

Proof that the difference of two positively squared integers never equals 1

This type of question is usually solved by a proof by contradiction, however I believe I have a direct proof of it, and I would like to know if its correct. Problem : Prove that there does not exist ...
1
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1answer
42 views

Proving Equality of the Induced Matrix Norm

I need to prove that the induced matrix norm satisfies $$\|A\| = \max_{\|x\| = 1} \|Ax\|$$ Here's what I've done so far, and I'm not sure how to make the connection. By definition, $$\|A\| = ...
0
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1answer
34 views

Proof that the first pivots in matrices $A, B$ will be in the same column.

This is a part of the proof that the reduced row echelon form of a matrix $M$ is unique, so please consider that when answering this question. Now what I need to prove is this: Suppose matrices ...
0
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1answer
13 views

Question about invariants.

There is a list of $n$ numbers. We pick any two numbers, $u$ and $v$ and replace them by $uv + u + v$. Does the final answer after $n-1$ operations, depend on the initial choice. I noticed that if ...
0
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0answers
15 views

Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $

I am trying to formally prove: $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $ where n is an integer, and x and y are natural numbers. It is obvious that, when $\frac xy$ is ...
4
votes
3answers
170 views

Steps to prove or disprove if two rings are isomorphic

So i'm struggling on how to prove if two rings are not isomorphic to one another. My professor told me that if a ring is not isomorphic to another, the best way to prove that this is true is to find a ...
1
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3answers
47 views

Question about a specific part of proving $\sqrt 7$ is irrational

I have a question that wants me to prove that the square root of $7$ is irrational. So I know we need to use proof by contradiction, then $7 = \frac{a^2}{b^2}$ where $a$ and $b$ are coprime. Then $a^2 ...
0
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1answer
35 views

Need help with proofs using axioms only [duplicate]

Prove if $p,q∈R$ and $pq>0$ then either $p>0$ and $q>0$, or, $p<0$ and $q<0$ using only the field axioms. I have no idea how to do this using only the field axioms. Seems pretty ...