For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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Number of distinct equivalence classes of $\mathbb Z_n$ of the “ associate ” equivalence relation

Define an equivalence relation on $\mathbb Z_n$ as : For $a,b \in \mathbb Z_n $ , $a\sim b$ iff $\exists k \in U_n=\mathbb Z_n^{\times}$ such that $a=kb$ (i.e. $a,b$ are related if they are "...
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1answer
31 views

Prove that $R\cap S$ is symmetric, transitive, and anti-symmetric.

If you can confirm these are done correctly or offer another way to do so I would greatly appreciate it. Also how would you go about proving $R\cap S$ is reflexive? What assumption if any would be ...
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0answers
13 views

Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding $2$ to both sides of an equation results in an equivalent ...
1
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0answers
41 views

Circles and generic implicit functions

I have some problems understanding circles. $x^2+y^2 = 1$ is a circle. It defines equivalence class where all (x,y) points belonging to the circle are in the same equivalence class. $(\cos a, \sin a)$...
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1answer
19 views

Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
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2answers
53 views

Equivalence relations and commutative diagrams

Let $\sim$ and $\dot\sim$ be equivanlence relations on the sets X and Y respectively. Suppose $f \in Y^X$ is such that $x \sim y$ implies $f(x) \dot\sim f(y)$ for all $x,y \in X$. Prove that there is ...
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0answers
19 views

How can i prove this equivalence relation problem when R isn't defined?

Problem: Prove that if $R\subset A\times A$, and $R\circ R^{-1}\circ R=R$. Then $R^{-1}\circ R$ is a equivalence relation. in $D(R)$ I have nowhere to take the properties i need from... what do I do? ...
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1answer
22 views

How can i prove that if $x_0$ is a solution then $[x_0]$ is unique?

$4x\equiv10\pmod6$ I'm not sure what they asking when they say that the equivalence relation of a solution is unique. Also I was able to find the solution -5 with euclids algorithm, is there a more ...
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2answers
55 views

We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that ...
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2answers
29 views

infinite equivalence classes

How would you prove that this relation $$(a,b)R(c,d)$$ if and only if $$a+d=b+c$$ has infinite equivalence classes if it is defined in a set with only non negative integers? I've already proved that ...
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22 views

Show that $E / \sim $ is an affine space over $V/U$

Let $E$ be an affine space over the vector space $V$, and let $U \subseteq V$ be a vector subspace. We define the equivalence relation $$P \sim Q := \exists v \in U \text{ such that } P = Q + v$$ on $...
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1answer
21 views

Proof about set of representatives

Let $X = \left\{(m, n)\in \mathbb{Z}\times\mathbb{Z}, n \neq 0\right\}$. Define a relation $\sim$ on $X$ by $(k, l) \sim (m, n)$ if $kn = lm$. Prove that $\left\{(m, n)|m \in \mathbb{Z}, n \in \...
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1answer
25 views

what is the complete set of representatives of an equivalence class?

I have been researching the topic, but I can't find anything that explains specifically and in detail what it is. I just find a bunch of exercises about the topic.
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17 views

Differentiating between equivalence relations and partitions

What is the purpose of defining equivalence relations and partitions separately when they are the same? Are they used for different purposes?
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1answer
36 views

Proof the existence of a certain bilinear form on the vector space V/T

Let $\mathbf V$ be a vector space (over a field $\mathbf K$) together with a symmetric bilinear form <-,->, and let $\mathbf T $ $\subseteq$ $\mathbf V$ be the orthogonal complement (since I'm not ...
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1answer
41 views

Prove that $\mathbb{|Q|} = \mathbb{|Q \times Q|} $ [duplicate]

This question exists, but both cases have a specific answer for the OP's situation. I do not know how to prove that $\mathbb{Q}$ is countable. Questions I am referring to: Prove that $\mathbb{|Q| = |...
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1answer
40 views

An uncountable chain of equivalence relations

First, an example: We know that, for two real valued, Lebesgue-integrable functions, the relation "equals almost everywhere" is an equivalence relation. In particular, if $f_0$ is Lebesgue-integrable, ...
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1answer
25 views

Equivalence relations proof example?

Let $A$ = {$a,b,c$}. Give an example of a relation on $A$ that is anti-symmetric, reflexive on $A$ and symmetric. The first thing that one must do to proceed with this question is to first define ...
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1answer
21 views

Fewest number of possible ordered pairs in a relation

There are many different equivalence relation possible on the set $A = \{a, b, c, d\}.$ For example, here are just two different ones: (a) $E_1 = \{(a, a), (b, b), (c, c), (d, d), (a, c), (c, a), (b, ...
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3answers
20 views

Defining a relation to a set

I have a homework question that asks me to define a relation A2 on $Z$ which is an equivalence relation containing three equivalence classes. $$Z = \{a, b, c, d, e\}$$ I understand what equivalence ...
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17 views

Technical Language Usage: Verb for an Equivalence Relationship “Forgetting” an Attribute that is “Modded Away”

This question is one of English usage, but I'm sure only mathematicians can answer it for me. I want to say in a technical report that an attribute is "ignored" or "forgotten" by an equivalence ...
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1answer
26 views

Equivalence relation and class, Proof.

The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$ (a) Prove that the the relation $P$ is an equivalence relation. (b) Describe the equivalence class of $3$ and $0$. In order to ...
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4answers
482 views

Definition of smallest equivalence relation

I came across the term 'smallest equivalence relation' in the course of a proof I was working on. I have never thought about ordering relations. I googled the term and checked stackexchange and couldn'...
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3answers
99 views

Quotient of a graph?

I want to understand quotient of a graph (also called quotient graph), my teacher says that the terms quotient of a graph and a modulo of a graph should be synonyms (even though modulo of a graph ...
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2answers
10 views

Find total number of relations that are equivalence as well as partial order set

Find total number of relations that are equivalence as well as partial order set. Assume set contains total $n$ elements. My attempt: As equivalence relation has property reflexive, symmetric and ...
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1answer
19 views

Constructing bijection from set of equivalence classes to another set

Suppose $f:A \to B$ is surjective. Define a relation on $A$ by setting $x\sim y$ if $f(x) = f(y)$. It is clear that $\sim$ is an equivalence relation on $A$. Let $\mathcal{E}$ be the set of ...
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36 views

Restriction to equivalence relation is equivalence relation

Let $\mathcal{R}$ be relation on $A$ and $A_0 \subseteq A$. The $\mathbf{restriction}$ of $\mathcal{R}$ to $A_0$ is defined to be the relation $\mathcal{R} \cap (A_0 \times A_0) $. $\mathbf{...
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39 views

Relations on equivalence classes

To be short, I will abstract a bit from my particular problem. Let $S$ be a set and $\sim$ be an equivalence relation, defined on that set. Let $R \subseteq (S/\sim) \times (S/\sim)$ be a relation ...
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2answers
58 views

Prove that $[a]=[b]$ iff $a\sim b$.

If $[a]=[b]$ then $a$ is in $[b]$ and $b$ is in $[a]$. If $a \sim b$, then $[a]$ is a subset of $[b]$ and $[b]$ is a subset of $[a]$. Then using the equivalence properties and showing that there is a ...
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2answers
24 views

Prove that $aRb$ if $a = 2^kb$ is an equivalence relation.

Let $R$ be a relation on the set of integers given by $aRb$ if $a = 2^kb$, for some integer $k$. show that $R$ is an equivalence relation. I don't understand how it will be equivalence. Is it the ...
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2answers
73 views

Elementary math proof

Let $\sigma$ : $\mathbb{Z}_{11} \to \mathbb{Z}_{11}$ be given by $\sigma([a]) = [5a + 3]$. Prove that $\sigma$ is bijective. Approach It has to be one to one and onto so It is one to one if $\sigma([...
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1answer
29 views

Transivity / Binary relation? [closed]

Discuss the Transitivity of Binary Relations $\mathcal{S} $ $a$ on $\Bbb R $ defined by $a (x, y)$ $\in \Bbb R^2 $--> $x \leq ay$ ( for some a $ \in \Bbb R$ ) I have this assignment about ...
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1answer
28 views

Let R be the relation on the set of ordered pairs of positive integers, Z+ × Z+, such that (a, b)R(c, d) if and only if ad = bc.

(For instance, (2, 4)R(6, 12) since 2·12 = 4·6.) Show that R is an equivalence relation. I was tasked to show that the sets is an equivalence relation if the three conditions Reflexive, symmetric and ...
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1answer
37 views

is this an equivalence relation? by Reflexive, Symmetric and Transitive.

i. {(a, b) : a and b have met} ii. {(a, b)} : a and b speak a common language i) Reflexive: yes Symmetric: yes Transitive: No, if a met b and b met a then a does not met c. ii) Reflexive: yes ...
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2answers
24 views

How to prove equivalence relation is disjoint?

I know how to prove when the equivalence are not disjoint, thus $[a]=[b]$. I see that the proof works for proving a equivalence relation is disjoint, but I don't get it. Can someone explain it to me? ...
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1answer
26 views

$X$ hausdorff and $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ is closed implies quotient map is open.

Let $∼$ be an equivalence relation on a topological space X. $\ Y = X/∼ $ equipped with the quotient topology. How to show that if X is Hausdorff and the set $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ ...
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3answers
62 views

How many equivalence relation can be defined on a set of $5$?

The question is how many equivalence relation can be defined on a set of $5$? I think this is asking how many different ways can we partition a set of $5$, right? So the answer is $1$ way: $$\{1\},\...
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3answers
24 views

Prove transitivity or not of some relation

I'm trying to prove if this equation is an equivalence relation or not. $R =\{(x,y) \in N\times N: \mbox{There exist }m,n \in N\mbox{ such that } x^m = y^n\}$ It's relatively easy to prove both ...
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1answer
27 views

Norms Equivalence over $\mathbb R^n$

Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequence in $\mathbb R^n$ is Cauchy under the $\|\cdot\|_2$ norm if and only if the sequance is Cauchy under the $\|\cdot\|$. Prove that a ...
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1answer
27 views

Find the multiplicative inverses of each nonzero element of the field $Z/(5), Z/(11), Z/(17).$

For $Z/(5)$, I figured that $[4]$ is a class that has an inverse of its own since $4 \equiv -1 (mod 5)$. Is that correct? Then I tried figuring that $[2]$ is also an inverse of its own since $2 \equiv ...
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2answers
32 views

Proving equivalence classes for a equivalence relation

I am having a bit of trouble trouble understanding how to start problems such as this one. I feel like I am given information that I understand separately but I can't seem to figure out how to they ...
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1answer
35 views

Why is this function well definded?

I have to take a look at the relation of $ x \sim y: \Leftrightarrow \exists k \in \mathbb{Z} : x-y=5k$ in $\mathbb{Z}$. I have no idea how to show at $\mathbb{Z} / \sim$ that the addition is well ...
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1answer
23 views

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function …

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function $f: A \to \Bbb N$ such that $\forall a, b \in A$ the following is true: $$a \mathcal R b \iff f(a) = ...
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40 views

How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
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1answer
64 views

Does the theory of equivalence relations have quantifier elimination?

I am aware that the theory of equivalence relations with infinitely many classes, all of which infinite, has quantifier elimination, as can be seen from the answer to this question. However, does the ...
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1answer
21 views

Solution check. How many relations of equivalence $R$ are there in $\Bbb N$ that verify silmultaneously the following properties.

I'm revising some old psets while preparing an exam and going through some things that I left unverified. How many relations of equivalence $R$ are there in $\Bbb N$ that verify silmultaneously the ...
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51 views

Prove that $(H,\circ)$ is a subgroup of the group $(G, \circ)$

Question: Let $(G, \circ)$ be a group and $H$ be a non-empty subset of $G$. A relation $\rho$ defined on $G$ by $$a\,\rho\ b\quad \text{if and only if}\quad a\circ b^{-1}\in H$$ for $a,b\in G$, is an ...
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21 views

How to phrase a proof of a equivalence relation of bijection

Define $\sim_\mathrm{bi}$ by $$\sim_\mathrm{bi} = \{(S_1,S_2)\mid \text{there is a bijection } f:S_1 \to S_2\}$$ for $S_1,S_2 \subseteq \mathbb{N}$ My proof comes as: In order to prove that $\sim_\...
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1answer
33 views

Count a Partial Equivalence Relations on a set

A Partial Equivalence Relation is a relation that is symmetric and transitive, and to count the number of equivalence relations on a set exists Bell Numbers, my question is How I can count the number ...
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1answer
69 views

Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...