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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Graduate Real Analysis research based question- Completion of the real numbers

I am currently a graduate student taking a real analysis independent study class. This is my first time taking real analysis, as I did not take it as an undergraduate. I am working on a research ...
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combine equal pairs into equivalence classes

Given a set $S$ of values, and a set $P \subset S \times S$ of pairwise equivalences, what is an algorithm for partitioning $S$ into equivalence classes? $P$ is guaranteed to be an equivalence ...
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Question about Equivalence Relations [closed]

A partition of the set X is a set of nonempty subsets of X such that every memeber of X is a member of one and only one subset. For example, the set of sets {{1, 3, 5}, {2, 4, 6}} is a partition of ...
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Subtlety in generated equivalence relations [duplicate]

I want to share my efforts on a (standard) exercise on generated equivalence relations, that I (surprisingly) struggled with at first. I took the exercise out of Lee's "Topological Manifolds"...
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What set theory results if (=)/2 would be a PER?

Assume we replace FOL= as the logical ground for ZFC by FOL=', where =' is only a partial equivalence relation (PER). Namely (=')/2 would be a relation which satisfies: ...
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Equivalence of full and sequential minimization

I would like to show that two problems are equivalent, so that by solving one of them I get the optimal solution to the second. Formally, I'm trying to prove that: $$ \min_{\mathbf{x,y}\in\mathbf{X\...
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Equivalence class for a determinant

Show that the relation on the set of all 2 x 2 matrices defined by A ~ B if detA=det B is an equivalence relation. Describe the equivalence class. I have determined that it is an equivalence relation. ...
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1answer
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Number of equivalence relation on $A=\{1,2,…,11\}$ such that $|A\setminus R| = 2$.

I am still trying to understand the question, I know that if a relation is reflexive (and equivalence relation is surely reflexive), then for every $a\in A$ , $(a,a)\in R$. But that means every ...
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1answer
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symmetric relation definitions

Definition One: A relation over a set $X$ is symmetric if for all $a,b$ $\in X$, $(a,b)\in R$ if and only if $(b,a)\in R$. Definition Two: A relation over a set $X$ is symmetric if for all $a,b$ $\in ...
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Let $R$ be an equivalence relation on $A$…

knowing that in $A$ is more than three elements. we define $R^c=A\times A \setminus R$, then which of these statements is surely correct? a) if $R^c$ is transitive, then $R$ has only one equivalence ...
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Confusion on why 2 equivalence classes are either equal or disjoint

So I've just started trying to teach myself some topology and in the book I'm reading there is a proof that 2 equivalence classes are either equal or disjoint. However I'm a bit confused on why the ...
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Proving that $\operatorname{rank}(\mathbf{A}) = 1$ if and only if $\mathbf{A}$ is the product of a non-zero column vector and a non-zero row vector

$\newcommand{\mn}[1]{\mathbf{#1}}$ $\newcommand{\vn}[1]{\mathbf{#1}}$ $\newcommand{\rk}[1]{\operatorname{rank}(#1)}$ $\newcommand{\t}{\mathrm{T}}$ As a beginner in linear algebra, I encountered an ...
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1answer
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Particular proof of the fundamental theorem of equivalence relations

I have some trouble understanding the following proof. After doing some research, I realized that this is (probably) the Fundamental Theorem of equivalence relations. I didn't understand the proofs I ...
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Find all the partition class sets of Z = {1, 2, 3} of the equivalence relation A R B = A ∪ Y = Y ∪ B.

We have partition class Z = {1, 2, 3}. And the task is to find all the sets of partition class of this equivalence R. And the relation is on the sets X = {1, 2, 3, 4, 5} and Y = {2, 4, 5} on the power ...
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how do you turn a set $(2z-1)$, given the partition into a relation

So this is just for my understanding, I'd like to know whether say if I'm given the set $S=2z-1$ , for some $z\in\Bbb{Z}$ does this mean the relation is $$R=\left\{\left((2x-1),(2y-1)\right) \in\Bbb{Z}...
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Equivalence classes of a kernel

Let $f$ be a function with domain $A$ and codomain $B$. Consider the relation $K \subseteq A \times A$ defined on the domain of $f$ by $(x, y) \in K$ if and only if $f(x) = f(y)$. The relation $K$ is ...
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Quotient space Hausdorff/$T_2$?

Let $n$ natural number and $X = [0,1] \times \{1,...,n\}$ with the topology $\tau \times \tau_D$, where $\tau$ is the usual topology of $\mathbb{R}$ induced by $[0,1]$ and $\tau_D$ is the discrete ...
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Proof that there is exactly one equivalence relation that forms a partition

So I've just started trying to teach myself some topology and i cant quite grasp how an equivalence class forms a partition more specifically i don't understand the proof that there is exactly one ...
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Where is the equivalence of the relation used in the construction of the numbers?

Given $\mathbb N=\{0,1,2,...\}$, the set of integers is defined by $$\mathbb Z=\Big\{\{(x,y)\mid (x,y)R(a,b)\}\mid (a,b)\in\mathbb N\times \mathbb N\Big\},$$ where $R$ is the binary relation defined ...
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Let $R$ be an equivalence relation on $X$ and $S$ an equivalence relation on $X/R$…

Let $R$ be an equivalence relation on $X$ and $S$ an equivalence relation on $X/R$. Find an equivalence relation $T$ on $X$ such that $(X/R)/S$ is in one-to-one correspondence with $X/T$ under the ...
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Calculating the distance between two objects with known heights in an image in which forced perspective appears to make them the same height?

I have a photograph containing two objects with known dimensions. Object A, in the foreground, is 23 inches tall. Object B, in the background, is 30 inches tall. In the photograph, however, both ...
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Question about proving ceiling function equivalence relation

I read a proof about ceiling function equivalence relation as linked here. The theorem to prove is: Let $\mathcal R$ be the relation defined on $\mathbb{R}$ such that: $\forall x, y \in \mathbb{R}: \...
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Existence proof for equivalence relation

Suppose an equivalence relation is defined as: for $a, b \in \mathbb{R}$, $a R b \iff a - b = n/m$, where we have $n, 0 \neq m \in \mathbb{Z}$. For any natural number $N$ and any $a\in \mathbb{R}$, ...
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38 views

Equivalence relation existence

Suppose an equivalence relation is defined as: for $a, b \in \mathbb{R}$, $a R b \iff a - b = n/m$, where we have $n, 0 \neq m \in \mathbb{Z}$. For any natural number $N$ and any $a\in \mathbb{R}$, ...
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1answer
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Rigorous proof for Theorem 7.7 in Loring Tu

Theorem 7.7 is stated and proved as follows in Loring Tu's An Introduction to manifolds: However, it seems that the deduction which involves Figure 7.4 in the proof is not so rigorous, so I want to ...
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1answer
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Relation of equivalence and equivalence class

(ZFC)Prove that the two conditions are equivalent: S/~≤T & T≤S ∃f (f:S↠T & ∀s1∀s2(f(s1)=f(s2)⇒s1~s2)). My attempt: I start with the first, trying to prove the second. S/~ is defined as the ...
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1answer
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Cardinality of a quotient set $\mathbb{N}^\mathbb{N}/\sim$

I'm having trouble with the following task: On the set of all functions $\mathbb{N}^\mathbb{N}$ an equivalence relation $\sim$ is defined in the following way: $f \sim g \iff \forall n \in \mathbb{N}. ...
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Example of equivalence relations $S$ and $R$ such that $SR$ is an equivalence relation while $R \cup S$ isn't

I need to give an example of equivalence relations $S$ and $R$ such that $SR$ is an equivalence relation while $R \cup S$ isn't. I've already tried to find such relations on small sets of natural ...
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Action on orbits defined by another action

Let $(G,\Omega)$ be a transitive action of $G$ on $\Omega$. Let now $H\le G$ be a subgroup of $G$ and define $\Omega/H=\{\omega\cdot H|\omega\in\Omega\}$ as the set of the orbits generated by the ...
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An $\omega$-stable theory that is not uncountably categorical

Morley's theorem states that a countable complete theory that's $\omega$-stable and has no Vaughtian pair is uncountably categorical. From the statement, I guess the "no Vaughtian pair" ...
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How many equivalence classes are there

$E={1,2,3,4,5,6,7,8}$ Defines the product set E × E the relation R: $(p, q) R (p_0, q_0) $if $ p-p_0$ even and $q-q_0 $divisible by 3 Question :How many equivalence classes are there My attempt : $p-...
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1answer
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How do I prove the set of the equivalence classes of $R$ has the same cardinality as the set of all finite sets of primes? [closed]

I have been stuck on this for a while now. Given the equivalence relation $R$ over $\mathbb{Z^+}: aRb \leftrightarrow \exists q \in \mathbb{Q}(\frac{a}{b} = q^2) $ how does one prove that the set of ...
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1answer
51 views

Let S be a subset of $\mathbb{C}$. If $z,w \in S$, define $z ∼ w$ if and only if there is a path from z to w. Show that ∼ is an equivalence relation.

Let S be a subset of $\mathbb{C}$. If $z,w \in S$, define $z ∼ w$ if and only if there is a path from z to w. Show that ∼ is an equivalence relation. Definition of path: A path in $\mathbb{C}$ is a ...
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Correct usage of $\equiv$ and $\doteq$?

I defined two objects, one at an initial time $t_0$, called $A(t_0) = \{ x \; | \phi(t_0, \,x)\}$ with $\phi$ a binary predicate, and a second object defined for some time $t > t_0$, called $A(t) = ...
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1answer
34 views

Material biconditional and substitution

Does the material biconditional have the substitution property? As an example, if I was able to derive $A \leftrightarrow B$ during an argument, may I substitute $A$ for $B$ (or vice-versa) in any ...
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Equivalence class for the following relation

A relation of $\mathbb{R}$ is defined as $a\sim b : a^4-b^2=b^4-a^2$ Show that $\sim$ is equivalence relation (I have done this part) Determine the equivalence class $[-1]_\sim$ Prove or disprove: ...
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How many different equivalence relations with exactly two different equivalence classes are there on a set with $n$ elements

I came across with this topic. It looks straight forward for $5$ elements, but what if I want to find how many different equivalence relations with exactly two different equivalence classes are there ...
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Big Theta as an Equivalence Relation

Forgive me as this is a question inspired by my first computer science algorithms class. I am a math major so when I learned of the big-theta operation on the space of continuous functions I began to ...
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1answer
56 views

Is R is an equivalence relation?

Hello so I'm trying to make an exercise, but it makes no sense to me it goes as follows. $Fun(X,Y)$ is the set of all functions $f:X\rightarrow Y$. We define a relation $R$ on $Fun(X,Y)$ by saying ...
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1answer
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Solution verification about equivalence classes.

I'm absolutely not sure about the notations and the correct terminology and I was wondering if everything checks out.. Question 5 a. $4^{2\ }-1$ = 15 Therefore we have 15 different partitions. b. The ...
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Proof regarding quotient sets and functions.

Okay so I was kind of winging it, especially in the second direction. I wanted to make sure what I wrote is not complete nonsense.. Assume $f$ is injective. Since $R$ is an equivalent relation, it ...
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1answer
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What is nedeed to prove that the tangent space on a manifold is a vector space? [duplicate]

I am currently working with the definition of tangent vectors being equivalence classes of curves. So $v =[\gamma]$ and $w=[\sigma]$ where $v,w$ are the vectors. I want to prove that the sum of this ...
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2answers
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How to prove if the relation R is an equivalence relation?

Hello so I stumbled upon an exercise where I need to prove that R is an equivalence relation, normally I have no problem doing this, but with this exercise I have absolutely no clue where to begin. I ...
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1answer
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Question about equivalent classes.

I have no idea what going on in this question, nor how to approach it. From messing around with this question I found 14 pairs (i.e. (1,2),(2,1) is considered one pair) that I can add to the original ...
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Proving a statement about equivalence relations.

Am I doing it correctly? I am not sure about my notations and the way I explain things as this topic is very new to me.. Question 3 a. True $E_{1}∩E_{2}$ Reflexive: Since $E_{1}$ is an equivalent ...
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1answer
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Relations and quotient groups.

We've started studying relations and I have no idea what I'm doing. Is this supposed to be the way to represent my answers? a. The relation is all possible relations such that the sum of a,b is equal ...
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1answer
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Question about relations and equivalence classes.

Hi so we've started learning about relations and I'm completely lost.. I don't get how to represent the equivalence classes and I'm not even sure the way I've proved the equivalence relation is ...
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1answer
54 views

Equivalence Relation question - $(a,b)S(c,d) \iff a-b=c-d$ such that $(a,b),(c,d) \in\ \mathbb{N}\times\mathbb{N}$

The relation $(a,b)S(c,d) \iff a-b=c-d$ such that $(a,b),(c,d) \in \mathbb{N} \times \mathbb{N}$ Need to find equivalent set for ($6,6)$ and $(2,5)$ I found for $[(6,6)] = \{(c,d) \in\mathbb{N} \...
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Determining equivalence classes of a matrix relation.

Problem: Given the equivalence relation, defined on $\mathbb{R}^{n\times n}$: $A \sim B$ $\Leftrightarrow$ $\exists S,T \in GL_n(\mathbb{R}): S B T$ Show that there are n+1 equivalence classes of &...

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