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.

Filter by
Sorted by
Tagged with
1 vote
2 answers
20 views

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: $m^3 + n = n^3 + m \rightarrow n^3 + m = m^3 + n$ ...
user avatar
2 votes
0 answers
16 views

Help finding mistake in proof involving the quotient map.

Consider the plane $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ with the product topology which has basis consisting of all open squares of the form $$\tag{1} ]a,b[\ \times\ ]c,d[\ \subseteq\mathbb{R}\...
user avatar
  • 133
0 votes
1 answer
28 views

Proving that the quotient of a set by an equivalent relation is a partition

I need to show that the quotient of a set $S$ with respect to the equivalence relation $\sim$ is a partition of $S$. To show this, we will denote the quotient by $P_\sim.$ Note that $$ P_\sim = \{[a]_\...
user avatar
3 votes
1 answer
36 views

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 ...
user avatar
0 votes
0 answers
15 views

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 ...
user avatar
  • 1
-1 votes
2 answers
44 views

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 $G$? ...
user avatar
  • 1,035
0 votes
0 answers
37 views

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 ...
user avatar
  • 379
-1 votes
0 answers
47 views

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 ...
user avatar
2 votes
1 answer
49 views

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 ...
user avatar
0 votes
0 answers
36 views

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 $y\mathrel R x$.
user avatar
0 votes
0 answers
13 views

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 ...
user avatar
  • 37
-1 votes
0 answers
22 views

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 ...
user avatar
1 vote
0 answers
15 views

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 $L = {b^∗a^n | n ≡ 0 mod 5}$ ...
user avatar
1 vote
0 answers
31 views

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 ...
user avatar
  • 258
1 vote
1 answer
34 views

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 ...
user avatar
0 votes
0 answers
33 views

Proving Inequality Operation is Well-Defined

Preliminary information: Let $\sim$ be a relation on $\mathbb{N^2}$ defined by $(a,b)\sim(c,d)$ if $a+d=b+c$. I am trying to prove that $\le$ is well-defined. Define $[(a,b)]\le[(c,d)]$ if $a+c \le b +...
user avatar
1 vote
1 answer
42 views

Help Finding the Size of a Quotient Set

I am having difficulties solving the following question. Given a finite set $X$ and $S \subseteq X$. $R$ is the equivalence relation over $P(X)$ defined as: $$(A, B)\in R \iff A \cup S = B \cup S$$ ...
user avatar
1 vote
0 answers
29 views

Finding equivalence class for $R_L$ of a regular language

Let $\Sigma = \{ a, b, c\}$, $$ L = \{w\in\Sigma^*\mid w \text{ starts with $ab$ and ends with $ab$}\}, $$ i.e. $L = ab(\varepsilon + (a+b)^*ab)=ab+ab(a+b)^*ab$. I need to find a regular expression ...
user avatar
5 votes
0 answers
64 views

Weaker assumptions to define an equivalence relation

Given a set $A$, the standard definition (at least Wikipedia's one) says that a binary relation $\sim$ on $A$ is an equivalence relation iff: $\forall a\in A \quad a\sim a\quad$ (reflexivity); $\...
user avatar
  • 2,811
1 vote
1 answer
20 views

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 ...
user avatar
  • 11
0 votes
0 answers
18 views

Proving Equivalance relation (transitivity)

$$relation = {(a, b) | b − 3 < a ∧ a < b + 3}$$ Unsure how to proof transitivity. Also is my reflexive and symetric proof strong enough? A = $\{\mathbb{Z}\}$ Reflexive: yes because $\forall x \...
user avatar
0 votes
1 answer
37 views

Reflexive relation on all integers

Let $R = \{(a,b)\mid a\cdot b=21\}$ be a relation on all integers. so $R = \{(1,21), (3,7), (7,3), (21,1)\}$ I fail to understand if the domain here is all integers or only A, the domain, $= \{1,3,7,...
user avatar
-1 votes
1 answer
35 views

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 ...
user avatar
0 votes
0 answers
43 views

category of monoids has all coequalizers as follows.

The following is from Problem 13 of Chapter 3 in Awodey's Category Theory. 13-Show that the category of monoids has all coequalizers as follows. Given any pair of monoid homomorphisms $f, g: M \...
user avatar
  • 4,081
1 vote
1 answer
37 views

Prove the function defined by $\pi(x) = [x]$ is a surjective map

I am working on a homework problem and would like some guidance/verification that I am on the right track. The problem statement is as follows: Prove that if $\sim$ is an equivalence relation of $X$, ...
user avatar
0 votes
1 answer
43 views

How is the equivalence relation for the cosets of a Polynomial Quotient Ring defined?

When you partition $Z$ into cosets of Equivalence Classes using say $5Z$, then each element of the group $Z/5Z$ is an equivalence class which is defined by the relation $p \equiv q \bmod 5$ where $p$ &...
user avatar
  • 359
-1 votes
1 answer
36 views

Prove that $R$ is an equivalence relation and that $C = X/R$

Consider a non empty set $C$ with non empty elements such that, for all $x$ and $y$ belonging to $C$ , if $x \neq y$ then $x \cap y = \emptyset$. Let $X = \bigcup C$ and define a relation $R$ as the ...
user avatar
0 votes
1 answer
40 views

Transitive binary Relation on sets

Consider a set , A = { 1 , 2 , 3 } Subset = { (1,3) , (1,2) } is a transitive relation . But I don't get that how it is transitive relation because there is no ...
user avatar
0 votes
1 answer
34 views

Define an equivalence relation on the set A={a,b,c,d} such that the equivalence classes are {a,b,c} and {c} [closed]

Define an equivalence relation on the set A={a,b,c,d} such that the equivalence classes are {a,b,c} and {c} Hello, sorry if the problem is very trivial, but I'm just learning and I have a question. ...
user avatar
0 votes
0 answers
27 views

Proving that the set of equivalence classes of $\mathbb{R}/\mathbb{Z}$ is $[0,1)$

Given $x,y \in \mathbb{R}$, we define $x \sim y$ if $x - y \in \mathbb{Z}$, so $\mathbb{R}/\mathbb{Z}$ is the set of equivalence classes, and I want to prove that the set of equivalence classes is the ...
user avatar
  • 1,184
0 votes
1 answer
33 views

For α and β ∈ Sn, define α ∼ β if there exists a σ ∈ $S_n$ such that $σασ^{−1}$ = β. Show that ∼ is an equivalence relation on $S_n$

My attempt is below. Could I please get feedback on it. I am not so sure that it is correct. Let α,β,σ ∈$S_n$. Since $S_n$ is a group, we know that it contains an identity. Let e be the identity. So, $...
user avatar
0 votes
0 answers
18 views

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 $T$...
user avatar
  • 51
2 votes
1 answer
22 views

How can we make the "time of first return" function well-defined? Context: dynamics and recurrence of measure-preserving systems

First, some preliminary definitions for those not familiar with the notations of the text on ergodic theory which I am following: Let $(X,\Sigma,\mu;\varphi)$ be a measure-preserving system - $\...
user avatar
  • 8,526
0 votes
1 answer
69 views

In (applied and pure) math study, do we only discuss and need material equivalence, not logical equivalence?

I am a mathematics major student and interested in logic. I have some questions, in math(both pure and applied aspects) study and research, do we clearly distinguish between logical equivalence and ...
user avatar
0 votes
1 answer
25 views

Equivalence of sup and Manhattan norm

I'd like to show the following statement to complete my proof: for every $n \in \mathbb{N}$ there exists a constant $C_{n}$ such that for every polynomial $f(t)=a_{0}+a_{1} t+\ldots+a_{n} t^{n}$ of ...
user avatar
3 votes
1 answer
87 views

How do I prove symmetry without a defined set?

I have a formula: ∀x,y, z(xRy ∧ xRz → yRz) If the formula holds for a relation, then the relation is Euclidean. If a relation is Euclidean and reflexive, what are the steps for proving it is also ...
user avatar
1 vote
0 answers
73 views

Prove that $\hspace{1mm} (\mathbb{R}^{n+1} \setminus\{0\})/{]0, \infty[}\cong\mathbb{S}^n$

I have been struggling to get a better grasp of the overall picture of what's going on throughout my solving process, perhaps, it's due a lack of a better understanding of some of the concepts ...
user avatar
1 vote
1 answer
52 views

Showing that equivalence class of path connected points is closed

Let $X\subseteq \mathbb{R}^{n} $ be open and $$[x]_{\sim} = \{y \in X \mid \text{there exists a continuous path from }x \text{ to }y \text{ in X}\}.$$ I want to show that $[x]_{\sim}$ is closed in/...
user avatar
  • 192
1 vote
0 answers
26 views

Equivalence relation between differential curves

I'm trying to solve the following exercise (number 2.59, from M. Abate, F. Tovena). Consider the fact that, as control-system engineer, I'm trying to learn differential geometry as self-taught. So, ...
user avatar
2 votes
1 answer
35 views

Sentence equivalent to $\bigwedge_{i=1}^ \infty \sigma_i$ without using infinite conjunctions

Let $\mathcal{L}$ be a language (with equality) containing a binary relation symbol $R$. Assume $R$ is interpreted as an equivalence relation (so it is reflexive, symmetric, and transitive) I see that ...
user avatar
1 vote
1 answer
34 views

I am very confused about this question can someone please help me solve it. Thanks!

Lucy works in the space $\mathbb{R}^n$ of vectors $x = [x_1, . . . , x_n]^T$ . Chris chosen a different basis and handles vectors $x’ = [x’_1 , . . . , x’_n ]^T$ , where $x$ and $x’$ are related by $...
user avatar
1 vote
3 answers
127 views

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 ...
user avatar
  • 11
0 votes
1 answer
92 views

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 ...
user avatar
0 votes
0 answers
52 views

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 ...
user avatar
  • 53
0 votes
0 answers
8 views

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. ...
user avatar
1 vote
1 answer
72 views

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 ...
user avatar
2 votes
1 answer
30 views

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}~(...
user avatar
  • 125
0 votes
0 answers
28 views

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 ...
user avatar
  • 2,302
0 votes
1 answer
41 views

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}}}}{\...
user avatar
  • 137
0 votes
1 answer
48 views

How to find implication classes in a graph?

I understood the general idea of comparability graphs and transitive orientation but just can't wrap my head around the implication classes. From Advanced Topics in Graph Algorithms - Ron Shamir. How ...
user avatar
  • 169

1
2 3 4 5
58