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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Proving an Equivalence Relation Question [duplicate]

Question: Let m ∈ ℤ+. Show that ≡m is an equivalence relation on ℤ. sorry, ℤ+ = positive integer
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Proving properties of relations on a power set.

Take $R$ to be the relation defined on $P(\{1, . . . , 100\})$ by $A \sim B$ if and only if $|A \cap B|$ is even. Firstly, am I right to think that for example, $|\{0\}\cap \{1\}| = |\{1\}| = 1$. And ...
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Equivalence relations, possible typo in textbook answer

9.78. Let $R_1$ and $R_2$ be equivalence relations on a nonempty set A. Prove or disprove the following: If $R_1$ ∩ $R_2$ is symmetric, then so are $R_1$ and $R_2$. The statement is false. Let A = {1,...
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Can equivalence relations be axiomatized using just one elementary sentence?

Equivalence relations are traditionally axiomatized by the Reflexivity, Symmetry, and Transitivity axioms. However, they can also be axiomatized by Reflexivity and Circularity. (Circularity is this ...
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Characterization of simple groups in terms of its conjugacy classes [closed]

Recently I have seen a post whose link is the following. I am not able to prove the first statement, namely, "A group $G$ is simple if and only if for any $1 \neq x \in G$, the conjugacy class of ...
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Can any subset of $\Bbb{N}$ be an equivalence class? [closed]

I am wondering if for any given $x \in P(\Bbb{N})- \{\emptyset\}$ we can find an equivalence relation such that it will have an equivalence class equal to $x$. Extend of this question is whether for ...
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Reference Request for Axiomatic/Algebraic Big $\mathcal{O}$ and Little $o$

I have seen the formal definitions of big $\mathcal{O}$ and little $o$, and do all right working with them. Still, I have some questions that a good reference might help clear up. In what level of ...
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Finding the unique representative that lies on the unit circle in an equivalence class of a given equivalence relation.

I am doing an exercise from a number theory textbook for practice and not sure how to approach this problem. Let $S:=(\Bbb R \times \Bbb R)\setminus{(0,0)}$. For $(x,y),(x',y') \in S$ let $(x,y)~\sim ...
3 votes
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Conjugacy classes of an element that are the same

Recently I have been studying the transfer homomorphism, and it came to mind that whether conjugacy class of an element with respect to some subgroup is the same as the original group. Namely, if $x \...
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Introductory equivalence relations xRY

I have a question about members/subsets. Let A be a nonempty set and let B be a subset of the power set $\mathcal{P} ({A})$ of A. Define a relation R from A to B by xRY if x ∈ Y. Give an example of ...
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Algorithm for sorting by equivalence relation

I hope the thread title isn't too strange, but I don't know better. My question seems a quite simple one. Having a set of objects I'm interested in the subsets that are pairwise equal. Example: A set ...
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create a equivalence relation with 3 equivalence classes [closed]

i need to find an equivalence relation R ⊆ EvenNumber × EvenNumber, witch contain 3 equivalence classes. i've try , R = (a,b) | a can be divided by b. But this have only 0 and all the other number as ...
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Understanding Reeb foliation

Reading this paragraph : https://en.wikipedia.org/wiki/Foliation#Submersions , I can picture the foliation of the band $[-1,1]\times \mathbb{R}$ induced by the map $f:[-1,1]\times \mathbb{R} \ni (x,y) ...
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How to show that R is an equivalence relation [duplicate]

If A be the set of Natural numbers (N) and let R be a relation on A × A such that (2a + 5b) is divisible by 7. Show that R is an equivalence relation
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Prove that a given set is subset of Permutation Group

For $l \in$ {$0, 1, 2, 3$} Let there be functions $\rho_{l}\: \mathbb{Z}_4 \rightarrow \mathbb{Z}_4$ and $\sigma_{l}\: \mathbb{Z}_4 \rightarrow \mathbb{Z}_4$ defined such that: $$ \rho_{l}([x]_4) \:= [...
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From equivalence relation to function

I need your help with the following task: I have to prove: If $ \sim $ is an equivalence relation on a set $ A $ and if $ C=\{[a ]_\sim \mid a \in A\} $ is the set of equivalence classes of $ \sim $, ...
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Determination of equivalence classes

I need your help with the following task: I have to determine the equivalence classes for the following equivalence relation $A = \{ (a,b) \in \mathbb{R} × \mathbb{R} |\quad |a| = |b| \quad \} $ I ...
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Distinct Equivalence Classes of Congruence Modulo 4

Let $A$ = $\{-4, -3, -2, -1, \space 0, \space1, \space2, \space3, \space4\}.$ $R$ is defined on $A$ as follows: For all $(m, n) \space\epsilon\space A,$ $\space \space mRn \Leftrightarrow 4 \space\...
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Quotient spaces-vector spaces

Let $S$ be a subspace of a vector space $V$. Prove that for any subspace $\overline{K}$ of $V/S$ there exists a subspace K of V such that $S\leq K$ and $\overline{K}= K$. I don't have a lot experience ...
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A particular relation $\star$ is defined on $\mathbb{N}$ by $x\star y$ iff $\exists k\in\mathbb{Z}$, $y=5^k x$. Is $\star$ an anti-symmetric relation?

Its asked for other things too like symmetry, etc. but I'm confused on how to go about anti-symmetry, if I pick $k = 0$ then $x = y$ and $y = x$, is this enough to say yes? On the other hand I can ...
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Topology of reals quotient by integers AND irrationals is indiscrete?

This question is similar to Consider $\Bbb R / \Bbb Q = \{ x+\Bbb Q : x \in \Bbb R \}$. Show that the quotient topology is the trivial one. but not the same. Consider the eq. relation on $\mathbb{R}$ ...
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Defining a Equality relationship when it already exists? I'm going crazy

(Equality Relation) There is one familiar relation between a set and itself that we consider every set $S$ mentioned in this text to possess: namely, the equality relation $=$ defined on a set $S$ by $...
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Trying to determine if this relation is reflexive, symmetric, antisymmetric and transitive

Let A be the set of all people who have ever lived. For x, y ∈ A, xRy if and only if x and y were born at least 30 days apart I want to determine whether the relation xRy is reflexive, transitive, ...
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How to prove the axioms of equivalence relation without using example but theoretically or using logic?

As we know , if $\sim$ is an Equivalence relation on a set $A$ then $\forall a, b, c\in A$ we have i) $a\sim a$ (Reflexive) ii) if $a\sim b$ then $b\sim a$ (symmetric) iii) if $a\sim b$ and $b\sim c$ ...
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Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects

Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
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Equivalence relations - Cardinality of a subset

Let $R$ be a subset of Z $*$ Z, the relation being congruent mod 4. How many equivalence relations $E$ are such that $R$ is a subset of $E$ and $E$ is a subset of Z $*$ Z. I know that (mod 4) has 4 ...
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Are two quadratic forms equivalent over $\mathbb{Q}$

Are the quadratic forms $q_1$ and $q_2$ equivalent over $\mathbb{Q}$? $q_1(x, y)=x^2+y^2 \text{ and } q_2(x, y)=x^2+3 y^2$ $q_1(x, y)=x^2+y^2 \text{ and } q_2(x, y)=-x^2-y^2$ $q_1(x, y)=x^2+y^2 \text{...
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Intersections between equivalence relations

Suppose R and S are equivalence relations on set A and B respectively. Then if I want to prove that $R \cap S$ is a relation on the set $A \cap B$, can I assume that $A \cap B $ is non-empty?
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Equivalence Relations from $\mathbb Z \to\mathbb Z$

The question is, "Which of these relations on the set of all functions from $\mathbb Z$ to $\mathbb Z$ are equivalence relations." The first relation to consider is $\{(f,g)|f(1)=g(1)\}$,it ...
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Relationship between the size of a set, size of an equivalence relation on that set, and the number of equivalence classes

I'm having trouble with this question on my discrete math HW: Let R be an equivalence relation defined on a finite set A, such that |A| = n and |R|= m. Let k be the number of equivalence classes of R. ...
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Show Proof Equivalance

Question: Assume $P$ is a partition on a set $A$ and define $$R_p=\{(a, b)\in A\times A : \exists U\in P (a\in U)\land (b\in U)\}$$ Show that $R_p$ is an equivalence relation on $A$. In this question, ...
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The continuity of a function defined on $\mathbb R/2\pi \mathbb Z$, and the distance for $\mathbb R/2\pi \mathbb Z.$

I'm handling a function $f$ from $\mathbb R/2\pi \mathbb Z$ to $\mathbb R$, $f : \mathbb R/2\pi \mathbb Z \to \mathbb R$. And I was told that the meaning of [$f$ is continuous at $a\in \mathbb R/2\pi ...
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Why are monadic categories over $\mathsf{Set}$ cocomplete?

$\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathcal{T}}$Given any monad $(\T,\eta,\mu)$ over $\set$, it is claimed that the Eilenberg-Moore category of algebras $\set^\T$ is cocomplete. More ...
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Question about ordering a set with an order relation

I have an order relation R that is defined with xRy if $|x|<|y|$ or $(|x|=|y|$ and $x<0)$ or $x=y$. I need to use this relation to order the set of integers. It was also stated that R is ...
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Number of conjugacy classes of a cyclic subgroup [duplicate]

Let $G=S_{15}$ (symmetric group on $15$ symbols). We say that any two subgroups $H$ and $K$ of $G$ are conjugate if $gHg^{-1}=K$ for some $g\in G$. This is an equivalence relation and the equivalence ...
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A function whose inverse relation is not a function

I need to come up with a function $f:X \to X$ where $X=\{1,2,3,4,5\}$ whose inverse relation is not the function $X \to X$. I’ve tried with for example $f(x)=|x|$, but this clearly doesn’t work.
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Disjoint equivalence classes of a set of matrices.

Let $M$ be the set of $m \times n$ real matrices. Consider the following equivalence relation: $A, B$ are equivalent if there exists a real invertible $m\times m$ matrix $P$ and a real invertible $n \...
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Is $R$ an equivalence relation or not? [closed]

Let $\Bbb N := \{1,2,3,\dots\}$ and a relation is defined in $\Bbb N \times \Bbb N$ as follows. $(a, b)$ is related to $(c, d)$ if and only if $ad=bc$ then show whether $R$ is an equivalence relation ...
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Floor function equivalence relation and partition

Could someone help me with this question? I would like to know if I did it right. Define the relation ∼ on $\mathbb{R}$ such that x ∼ y if and only if ⌊2x⌋ = ⌊2y⌋. 1) Prove that ∼ is an equivalent ...
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Symmetric property of an equivalence relation states that if $a$~$b$ then

Symmetric property of an equivalence relation states that if $a$~$b$ then $b$~$a$; Transitive property states that if $a$~$b$ and $b$~$c$ then $a$~$c$. What is wrong with the following proof that ...
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Let $(M, d)$ be a metric space. Set (the default bounded metric) $\bar d : M \times M \to R$ by $\bar d(x, y) =$min {$d(x, y), 1$}, Show that:...

Let $(M, d)$ be a metric space. Set (the default bounded metric) $\bar d : M \times M \to R$ by $\bar d(x, y) =$min {$d(x, y), 1$}. Show that: a) $\bar d$ is a metric in $M$. b) $d$ and $\bar d$ are ...
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Topology on $\mathbb{I}$

Let $\mathbb{I} = \mathbb{R}\backslash\mathbb{Q}$ be the set of irrational numbers. Define the equivalence relation $\sim$ on $\mathbb{I}$ such, that $\forall x,y\in\mathbb{I}:\big((x\sim y) \...
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Understanding the definition of cones in James Dugundji's.

James Dugundji defines (Topology, chap. VI definition 5.1) a cone in the following manner For any space $X$, the cone $TX$ over $X$ is the quotient space $(X\times I)/R$, where $R$ is the equivalence ...
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Which of the properties reflexive, symmetric, anti-symmetric and transitive does these relations have?

Which of the properties reflexive, symmetric, anti-symmetric and transitive have the relations $R$, $S$, and $T$ below? a) $X$ is the set of all functions $f: \mathbb{R} \setminus \{0\} \to \mathbb{R} ...
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Do functions defined on a torus have to be periodic?

Consider the unit torus $\mathbb T:=\mathbb R /\mathbb Z$. As far as I understand, the points of $\mathbb T$ are equivalence classes $[x]$ defined through the relation $x\sim x+k$, with $k \in \mathbb ...
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Cannot there be more Partitions or Equivalence Relations than the other? [duplicate]

The first assignment in an introductory Group Theory course, we're asked to review basic Set Theory and one of the questions came up which asks us to find the total number of possible equivalence ...
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Logical Consequence - Equivalent Assertions

I have the following slide in my notes and I'm having trouble understanding how the three assertions are equivalent. I understand to a degree how the 2nd and 3rd assertions are equivalent, but the ...
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2 answers
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Do equivalence classes partition all of the elements of the set that are in the relation or does it partition the entire set?

My discrete math textbook says that "The equivalence classes associated with an equivalence relation on a set A form a partition of A." However, I am not sure how this would hold for ...
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Relation of equivalency $f^n(x)=f^m(y)$

Lets consider a set $X$ and the relation $\Re$ defined in the following way: $x\Re y\Leftrightarrow \exists n,m\in \mathbb{N}, f^{n}(x)=f^{m}(y)$ $f^{n}=$identity if $n=0$ else $f^{n}=f•f^{n-1}$. Can ...
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Equivalence of atlases of vector bundles is an equivalence relation

I'm trying to show that equivalence between atlases of vector bundles is indeed an equivalence relation. Definitions Let $ V\subset \mathbb R^n $ be an open set and let $ F $ be a (real, finite ...

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