# Tagged Questions

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

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### Equivalence Relation Proof Question [duplicate]

Let $R$ be the relation on $N\times (N\setminus\{0\})$ defined by $((a, b),(c, d)) \in R$ if $ad = bc$. Prove that $R$ is an equivalence relation. I'm pretty confused with this problem, mainly ...
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### If a relation is built with $=$, is the relation always an equivalence relation?

$R \subset \Bbb R \times \Bbb R$ I have now encountered a couple of relations that have the following form: $$R=\{(a,b)\in \Bbb R\times \Bbb R\,:\,a^2 = b^2\}$$ They seem to be always equivalence ...
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### Is $R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $\}$ an equivalence relation?

$R \subset \Bbb N \times \Bbb N$ Is this an equivalence relation? $R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $\}$ I say it is not because $(a, a)$ is always $0$ which is ...
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### Is the relation $R$ on $\Bbb N$ given by $(a,b)\in R\iff a\mid b$ an equivalence relation?

$R \subset \Bbb N \times \Bbb N$ Is this an equivalence relation? $$R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,a\mid b\}$$ I would argue that it is reflexive because $a\mid a$, but it is not symmetric ...
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### How to prove equivalence of different definitions for compactness?

My workbook considers three different definitions for compactness in logic. It says that it can be shown that these are equivalent, but what would be a strategy to show this? I'm familiar with showing ...
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### Describe the equivalence classes generated by T

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$. Question Describe the equivalence relation T on the real line that is the intersection of all equivalence ...
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### Is $R$ an equivalence relation?

Let $X,Y$ be infinite sets. Define $F$ as $F=\{f:X\rightarrow Y\}$ . We define a binary relation $R$ on $F$: $fRg$ if there is no countable $S\subseteq X$ such that $\forall x\in S \ f(x)\neq g(x)$. ...
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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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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'...
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 ...