# 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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### Are $\phi^{'}_i,\tau_i$ the coresponding compatible functions for equivalence relations constructed in the directed limit of groups/rings.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, How to Prove it by: Dan Velleman, and An Invitation to Abstract Algebra by ...
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### Constructing compatible functions $\phi'_i,\phi'_j$ for equivalence relation $\sim$ for directed limit of directed systems of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
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### Set of equivalence classes as the image of a map

In the book I am reading it states that any map of sets $f\colon S \to T$ gives us an equivalence relation $\bar{S}$ defined by $a\sim b$ if and only if $f(a)=f(b)$. This all makes sense to me, but ...
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### Is this correct notation for set of equivalence class and quotient sets for directed limit of directed system of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
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### Are we assuming a relation to be transitive untill proven otherwise?

I do not know if the title would be correct title for the question but I think I am asking a valid question. While studying set theory and relations we were often asked about whether a relation $R$ is ...
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### How to describe the transitive closure of a relation in terms of the ground relation

Let $X$ be a non-empty set and $\equiv$ a relation on $X$, which is symmetric and reflexive but is not transitive. I know that there exists a transitive closure of $\equiv$, saying, $\equiv_{cl}$. But ...
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### My question even if any data set in integer doesn't satisfy the transitive we can genralize thats its not going to be transitive [closed]

Let R be a relation on ZZ defined by (a, b) * R(c, d) if and only if ad – bc is divisible by 5. Then R is
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### Coequalizer in the category of modules

I am trying to prove that the category of modules is cocomplete. It suffices to show that it has all coequalizers and coproducts. It's relatively easy to show that all coproducts exist, and I am left ...
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### Question regarding Transitivity of a Relation

Suppose we define a relation $R$ in the natural set $\mathbb N$ which says: $$(x,y)\in R\iff x^2-4xy+3y^2=0$$ and we would like to find which of the following properties does $R$ satisfy. My book ...
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### How much choice is needed to prove that $|G| \nless |G/ \sim|$ for any equivalence relation $\sim$ on set $G$?

$G/ \sim$ is the set of $\sim$-equivalence classes in $G$ and $|G/ \sim|$ is the cardinality of $G/ \sim$. $|A| \leq |B|$ means that there is an injective function from $A$ to $B$. $|A| < |B|$ ...
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### number of "equivalence relations" on a set with "n-elements"

I am trying to find a formula for number of equivalence relations on a set with n-elements however I am confused. I have already encountered the idea of "bell's number" and "Stirling ...
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### Relabel According to the Order of First Occurrence

Let $a\in\mathbb R^n$ be a tuple of length $n\in\mathbb Z_{>0}$. Let $X=\{a_i:1\le i\le n\}$ be the set of elements of $a$ for $x\in X$ let $$i(x)=\min\{j:a_j=x\}$$ be the first occurence of $x$ in ...
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### How change an non-equivalency relation to equivalency relation?

I am an engineer, so maybe this question is naive. I study equivalence relations and equivalence classes. An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. ...
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### Defining a measure on a group from a measure on the equivalence classes

Let $G$ be a group, and a $H$ a subgroup, and for $a,b \in G$ let $a\sim b$ if $aH = bH$. Suppose I have a measure $\mu$ on $G/\sim$, the left cosets of $H$, and suppose that $G$ is equipped with a ...
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### Let $B^2 :=\{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$. Show $B^2$ is equinumerous to $\mathbb{R}$

Let $B^2 :=\{(x,y,x) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$. Show $B^2$ is equinumerous to $\mathbb{R}$. I think there are a few ways to do this, probably some are easier, but I've committed to ...
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### Equivalence Relation Textbook Mistake

I found a question in my textbook but I think the answer provided is wrong. The question says: Let $S$ be a relation defined over $\mathbb R$ such that $(a,b) \in S \iff ab≥0$. Is $S$ equivalence? ...
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### If there is a bijection $F : A \mapsto A / R$, then $R = \{(x, y) \in A^2 : x = y\}$?

Assume $R$ is an equivalence relation over $A$ and there is a bijection between $A$ and $A / R$. Does this entail $R = \left\{ (x, y ) \in A^2 : x = y \right\}$? What I thought is the following. ...
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### equivalence classes vs partitions in imperfect-information games

I am reading Multiagent Systems by Yoav Shoham, and in chapter 5 about extensive games, the definition of imperfect-information games extends the perfect-information games definition with equivalence ...
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### Inverse images of sets versus inverse functions - should we use different notation?

I am reading the following paper Beloso-Herves, C. and Monteiro, P.K. Information and s-algebras. Economic Theory, 54(2): 405-418, 2013. and have a question of a technical nature which the authors ...
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### Equivalence relations, formal languages, and hypercube walks - what did I unleash on my students?

I teach a class in discrete mathematics and formal language theory. On my most recent final exam, I asked a question that involved a crossover between equivalence relations and formal languages. Here'...
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### Equivalence classes of $\mathbb{R}/\mathbb{Z}$.

I am trying to write down and prove a precise characterization of the equivalence classes of the relation on $\mathbb{R}$ defined by $x \sim y$ if and only if $x - y \in \mathbb{Z}$. What I've done so ...
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### Intuition on equivalence classes from group action on $a$ and index of the stabilizer of $a$ [duplicate]

I am currently reading Abstract Algebra by Dummit and Foote Chapter 4. I was trying to grasp the idea behind proposition 2, I could prove it but could not construct a geometric understanding of the ...
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### Is there any way to have a geometric, visual intuition of what this manifold could be?

Let's consider $H$ the space of $3\times3$ matrices with real coefficients of the form $$A =\begin{bmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ z & y & 1 \end{bmatrix}$$ with the ...
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### Trigonometric proof of the equivalence $\arctan [\frac {1} {2}] - \arccos [{\frac {1+3 \sqrt{3}}{2 \sqrt{10}}}] = \frac {\pi} {12}$

Solving this problem Two identical circles passing through each other's centres. Three parallel lines and two diagonal lines drawn as below. What is the value of the marked angle? I found that the ...
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### The equivalence relation is defined on a set of 9 elements and is negatively transitive relation. How many equivalence classes can be there?

I looked for answers a lot but mostly used the Bell number, which we haven't studied. And I have no idea how to do it with negatively transitive relations.. (Edit) Negatively transitive relation, ...
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### Proving that we can define a paradoxical set in terms of equidecomposability.

I'm doing an undergrad project on the Banach-Tarski paradox and I'm not convinced by the proof I have come up with for this, everything to do with the Banach-Tarski Paradox is new maths to me and so I ...
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My question is that suppose a set $A = \{1,2,3,4\}$ so, the identity relation can be $R_1 = \{(1,1), (2,2), (3,3), (4,4)\}$ but can it be $R_2 = \{(1,1), (2,2)\}$?