# Questions tagged [equivalence-relations]

For questions about relations that are reflexive, symmetric, and transitive. These are relations that model a sense of "equality" between elements of a set. Consider also using the (relation) tag.

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### Showing that $x^3 + y = y^3 + x$ is an equivalence relation

I am asked to prove that: $x^3 + y = y^3 + x$ is an equivalence relation. So far I have the following: Reflexive: $m^3 +m = m^3 +m$ Symmetric: $m^3 + n = n^3 + m \rightarrow n^3 + m = m^3 + n$ ...
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### Characterize open sets of this quotient topology.

Let $X$ the quotient space obtained from $\mathbb{R}\times\{0,1\}$ identifying $(x,0)\sim(x,1)$ if $|x|>1$. Which are the open sets of this quotient topology? First, I've made the next drawing to ...
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### Objects that change their equivalence class under some transformation

Suppose there is a set of objects on which we can define an equivalence relation. Under some transformations of the space on which the objects are defined, these objects may change their equivalence ...
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### When all commutant (centralizer) subgroups are abelian

I have seen the following problem in chapter 9 of Abstract Algebra by Dan Saracino: Let $G$ be a group and for $a,b \in G$ let $a\ R\ b$ mean that $ab=ba$. Must $R$ be an equivalence relation on $G$? ...
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### Intuition behind the definition of matrix similarity/equivalence?

Given two matrices $A$ and $B$, they are similar if: $$B=P^{-1}AP$$ Furthermore, if they are similar they are relative to the same linear transformation (equivalent). However the proof I've checked ...
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### Equivalence classes of positive definite binary quadratic forms [closed]

Find all equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$ with discriminant $D = −164$ under the action of the group $SL_2(\mathbb{Z})$. Can you find the ...
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### Prove that $x\mathrel Ry\iff4\mid(3x+y)$ is Transitive

Let $x,y,z \in\mathbb Z$ and suppose that $x\mathrel Ry$ and $y\mathrel Rz$. Therefore $4\mid(3x+y)$ and $4\mid(3y+z)$. So there exists $k, l \in\mathbb Z$ such that $4k=3x+y$ and $4l=3y+z$. Add these ...
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### Prove that $x\mathrel Ry\iff4\mid(3x+y)$ is symmetric [closed]

Does my proof look correct? Let $x, y \in\mathbb Z$ and suppose that $x\mathrel R y$. Thus $4\mid(3x + y)$. So $4\mid(3y + x)$. So $y\mathrel R x$.
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### Feedback on Equivalence Relations Proof from the Book of Proof

I wanted to get some feedback on a proof on equivalence relations in the Book of Proof. Suppose R is a reflexive and symmetric relation on a finite set A. Define a relation S on A by declaring xSy if ...
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### Relation of divisibility is antisymmetric relation [duplicate]

Antisymmetric relation : aRb and bRc implies a=b.....means we have to prove if a|b and b|a implies a=b According to divisibility : for a and b belongs to natural number a|b and b|a Implies b=am,a=bn ...
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### equivalence classes of a language via myhill nerode

My task is to find equivalence classes for different languages based on Myhill-Nerode. I'm having a hard time finding these equivalence classes; for example, the language $L = {b^∗a^n | n ≡ 0 mod 5}$ ...
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### Equivalence relation induced by a free action of an infinite countable group is aperiodic?

I am reading on how to calculate the cost of direct products in measurable group theory. In the proof by Gaboriau we have the following: Let $\Gamma, \Delta$ be infinite countable groups. We want to ...
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### About Quotient Space and Equivalence Classes

The following are the details of the problem: Let $M = \{m,a,t,h,f,u,n\}$ and $N = \{h,u,m,a,n\}$. For any $A,B \subseteq M$, $A \sim B \iff A \cap N = B \cap N$ where $\sim$ is an equivalence ...
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### Bijective function from relations of equivalency set to partitions set

I am proving that f: R->P, a function that associates a relation of equivalency to a partition is bijective. Let g: P->R. I showed that g(p=partition of X)={(a,b) belonging to X²| exists A ...
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### How do I prove that the following relation R is an equivalence relation?

I have the relation $R$ on $ℚ$, and $R$ is set to be the relation $R$ = {$(a,b): a - b ∈ ℤ$}. I am supposed to prove that the relation $R$ is an equivalence relation. I know I'm supposed to prove that ...
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### Can anybody point me to an existing statement or proof of this simple theorem about equivalence relations?

I'm trying to reduce a problem in computer science to something previously solved. My problem is about a set $S$ under a nice relation $\mathcal{E}$. The previously solved one is about sets $S$ and $T$...
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### Infinite groups: Does a^k=b^k imply a=b? Counter-examples?

Question in the end, but some background for the question: I'm reading Pinter's Book of Abstract Algebra (for fun, so not a course exercise) and Chapter 12 on equivalence classes exercise D.4 really ...
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### The world where every sequence is convergent

One day, I thougth "If every sequence is convergent, it is very nice." So I tried to define some equivalence class on the set of sequences. And I also tried to define operations such as ...
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### Fundamental group of $\Bbb{S}^1\times\Bbb{S}^1 /\sim$

Let $\Bbb{S}^1$ the unit sphere in $\Bbb{R}^2$ and $X=\Bbb{S}^1\times\Bbb{S}^1 /\sim\hspace{0.1cm}$, where $(p,q)\sim(q,p)$ for $(p,q)\in \Bbb{S}^1\times\Bbb{S}^1$. Compute $\pi_1(X)$. I'm trying to ...
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### How to determine equivalence from Westlake symmetric CI?

How can I know if A and B are bioequivalent via Westlake symmetric CI? I have 2 variable, and ran westlake CI. I got low and upper CI -4.858595058 , 4.85859505 . But I have no idea what this means. ...
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### Is this a valid justification that the relation $R$ is not transitive?

The question I was working on was asking to find whether the relation $R$ defined on $\mathbb{Z}$ by $a\,R\,b$ if $|a-b|\leq 2$ is reflexive, symmetric, and/or transitive and provide justification. I ...
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### Finding and drawing equivalence class with a binary set defined on RxR

In the question, it has been asked to find and draw the equivalence classes of the relation $∼$ on $(\Bbb R\times\Bbb R)\setminus\{(0, 0)\}$ which is defined as (x_1, x_2) \sim (y_1, y_2)~\text{if}~(...
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### Notation of Integers Modulo $n$ in regards to Equivalence Classes [duplicate]

Suppose I want to notate the integers modulo $n$. I used to use the notation $\mathbb{Z}_n$, however I recently learned that the notation $\mathbb{Z}/n\mathbb{Z}$ is preferred to represent the ...
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### Forms of Nelson Siegel formula

Are these two forms of Nelson Siegel formula equivalent? \$s_{m}(\beta )=\beta _{0}+\beta _{1}\frac{1-e^{\frac{-m}{\tau _{1}}}}{\frac{m}{\tau _{1}}}+\beta _{2}\left ( \frac{1-e^{\frac{-m}{\tau _{1}}}}{\...
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