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

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Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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1answer
743 views

Definition of “quotient set”

I search and search about quotient set and cannot figure out what it is. At the beginning, I thought it was the same as partitions, but now I'm confused. Can someone show some examples and explain?
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82 views

Binary relation, reflexive, symmetric and transitive

I have a question regarding an image. I'm currently studying binary relations and the following image confused me: What got me confused is that the page from which I got the link ...
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1answer
52 views

Homeomorphism are equivalence relations, so what are the equivalence classes?

Homeomorphisms are equivalence relations, so what are the equivalence classes for two Topological spaces $T_1, T_2$? Intuitively it seems like we might have the following equivalence classes - ...
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50 views

Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces

Let $X$ be a smooth projective surface and $D_1, D_2$ be two divisors on $X$. Is it true that $D_1$ is linearly equivalent to $D_2$ if and only if $D_1$ is algebraically equivalent to $D_2$?
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29 views

A partial order with more properties than would be expected

Consider the relation: $$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$ This is usually used for defining the (positive) fractions $\Bbb ...
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33 views

Can one talk about equivalences classes in a class(which might not be a set). [duplicate]

equivalence relation on a set breaks it into disjoint sets. does 'disjointness' make sense in classes. specifically, given a category, isomorphism classes of objects makes sense or not. can it be so ...
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110 views

Proving that $aRb \iff a^2-b^2=a-b$ is an equivalence relation

Could you help me with that, I don't know how to prove if the relation is an equivalence and the class of 5? On the set of integers, the relationship is defined by $aRb \iff a^2-b^2=a-b$. Find out ...
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30 views

Equivalence Relation with multiples

How can I prove the equivalence of this relation, and how can I calculate the equivalence class of (4,8)? On the set the relation R is definded by (a,b)R(c,d) ⇔ ad=bc. Find out if this is an ...
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1answer
25 views

Being smooth homotopic relation: proof

Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that ...
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1k views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
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22 views

Describe the equivalence classes in terms of familiar mathematical objects [duplicate]

Consider the equivalence relation $\sim$ on $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ defined by $(a,b) \sim (c,d)$ if $a \cdot d = b \cdot c$. Describe the equivalence classes in terms of ...
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554 views

Reflexive but not Transitive relation

What is an example of a relation $\mathscr{R}$ on a set $S$ such that $\mathscr{R}$ is reflexive but not transitive? Here is what I have come up with. Let $S = \mathbb{Z}$. Then let $\mathscr{R} = ...
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31 views

Equivalence Classes for 7 divides (x-y)

How do I find the distinct equivalence classes for the relation $(x,y)\in R$ if and only if $7$ divides $(x-y)$?
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52 views

Lifting an equivalence relation on elements to a relation on sets

If every element of $A$ is equivalent (by an equivalence relation $R$) to some element of $B$ $\forall x\in A\ldotp \exists y \in B\ldotp x\,R\,y$ and vice versa $\forall y\in B\ldotp ...
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2answers
47 views

Properties of bijections

If a bijection exists between set A={a1, a2, ...} and set B={b1, b2, ...} such that a1 maps to b1 and a2 maps to b2, etc., does this mean if we find a relationship R between a1 and b1 (i.e. f(b1) is ...
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74 views

How can we prove these equivalence relations [closed]

We have this relation $A= (\mathbb Z_{\geq0})\times(\mathbb Z_{\geq0})\times\ldots\times(\mathbb Z_{\geq0})$. And we have the relation $R$ on $A$ such that: $(a , b)R(c , d)\iff a+d=c+b$ and ...
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38 views

Find the equivalence classes

Prove or disprove: There is an equivalence relation $\sim$ on $\mathbb{Z}$ defined by $x \sim y$ if $x − y$ is even. What are the equivalence classes? I have proven that there is an equivalence ...
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1answer
17 views

Is there any special name for an algebraic structure (set, equivalence relation)?

I've seen the term "real multiset" but it doesn't seem to be very appropriate so i wonder whether there are any others. My second question is about multisets. In most sources i've seen one of the ...
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1answer
43 views

Is the property reflexive, symmetric, anti-symmetric, transitive, equivalence relation, partially ordered given the relation below?

I'm working on this and I'm supposed to figure out if the following properties apply to the below relations. Properties are: ...
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27 views

Question about partitions with a single element and equivalence relations

I couldn't find a formal definition of a partition but I found this picture on the Bell numbers wiki. You can see there are no partitions with a single element, it confused me, why a partition with ...
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37 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
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2answers
49 views

Finding the equivalence classes of a trigonometric relation

I have been asked to respond to the following: Define a binary relation R on $\mathbb{R}$ as ${\{(x, y) \in \mathbb{R} \times \mathbb{R} \mid \sin(x) = \sin(y)\}}$. Prove that R is an ...
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66 views

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$. Show that $R$ is an equivalence relation.

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$ a. Show that $R$ is an equivalence relation. My attempt: By definition 6.2.3 $R$ is an equivalence ...
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37 views

Divisibility: a partial order

Why is divisibility on the set of integers only a partial order and not a total order? I know total order requires the additional requirement of: For any $a,b \in S$, either $a\le b$ or $b\le a$. But ...
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2answers
46 views

Equivalence and Order Relations

I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation. Conjecture for which sets and this ...
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1answer
55 views

Equivalence Relation using Binary Operations.

Question: Let ∗ be a binary operation on a set A. Assume that ∗ is associative with identity. Let R be the relation on A defined on elements a,b ∈ R as follows: aRb if there exists an invertible ...
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54 views

Equivalence relations and equivalence classes

I dont know how to start this proof? Also, our professor did not explain equivalence classes fully so I am not understanding them very well.
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1answer
39 views

Show that this relation is an equivalence relation.

Given functions $$f_1 : A\to B$$and$$f_2 : A\to B,$$ let us write $f_1 \equiv f_2$ when there exist bijections $\alpha : A\to A$ and $\beta : B \to B$ such that $f_2(\alpha(a)) = \beta(f_1(a))$ for ...
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100 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
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65 views

For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$

For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one ...
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277 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
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61 views

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation. [duplicate]

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2}$ . Then $\equiv$ is an equivalence relation on $\mathbb{R}$ , there are infinitely many equivalence classes, one of them ...
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2answers
51 views

equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H $. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
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1answer
26 views

Topologically equivalent metric spaces is an equivalence relation

I'm trying to prove that topological equivalence is an equivalence relation. Reflexivity was easy, and I'm sure transitivity is too, but I'm stuck on symmetry. My book's definition is that a metric ...
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77 views

Improper integral calculation and nature

I would like to know how to calculate this integral n check its nature : $$ A= \int_1^\infty \frac{e^{-t}}{1+t^{2}} dt . $$ I think I figured it out , we can do it with Equivalence like this (since ...
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1answer
144 views

Picture - Equivalence Relation & Classes, Partitions, Quotient Set, & other related ideas

To get intuition for them and to remember them, I'd be grateful for a picture that combines and embodies the key definitions regarding Equivalence Relations & Classes, Quotient Sets, and ...
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1answer
29 views

Need help understanding transitive relations

My discrete math professor gave an example stating that the following relation is transitive, reflexive, symmetric, and antisymmetric. A = {a,b,c,d} R = {(a,a), (b,b), (c,c), (d,d)} I do not ...
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20 views

equivalence relation for $2a - b = 2c - d$ where $a,b,c,d$ are elements of $\mathbb{R}$

For $(a,b), (c,d)\in \mathbb{R}^2$ define $(a,b)\sim (c,d)$ to mean that $2a−b = 2c−d$. Prove that $\sim$ is an equivalence relation on $\mathbb{R}^2$. Reflexive: let $a$ be an element in ...
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1answer
27 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
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1answer
45 views

How many equivalence relations there are on a set with 7 elements with some conditions

Calculate how many equivalence relations there are on $\{1,2,3,4,5,6,7 \}$ that include the set $\{(2,2),(1,3),(3,6),(7,5)\}$ and are foreign to the set $\{(1,7),(4,7),(4,3)\}$. Well I first drew ...
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1answer
112 views

Equivalence relations and their class

I am trying to figure out how to prove that an equivalence relation of the relation x~y defined on Z <=> x-y is a multiple of 3. My attempt was: 1) Reflexive: x = x => x ~ x 2) Symmetric: x ~ y ...
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1answer
29 views

Find a subspace homeomorphic to the quotient.

I'm giving the equivalence relation $V\sim W$ when $\exists\lambda>0:\lambda V=W$. I have to prove that $\sim$ is an equivalence relation and find a subspace of $\mathbb{R}^2$ homeomorphic to the ...
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1answer
22 views

Equivalence relations proof

I need to prove that if $R_1$ and $R_2$ are equivalence relations on the set $A$, then $R_1\cap R_2$ is an equivalence relation. Problem is I dont know how. Please help!
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1answer
50 views

Discrete Math Proofs, Partial Orders and Equivalence Relations

I am horribly stuck on $3$ proofs for my discrete math class. Any help would be greatly appreciated. Prove that if $R$ is a partial order, then $R^{-1}$ is a partial order Prove that if $R_1$ and ...
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1answer
36 views

Prove a relation is a equivalence

Let $\sim$ be defined so that $a\sim b$ when $a+b$ is even. Is this an equivalence relation? Equivalence relations confuse me a lot, so any help is appreciated!
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1answer
37 views

Why does x ~ y <=> [x] = [y]

So I read in wikipedia that "It follows from the properties of an equivalence relation that $x \sim y$ $⟺$ $[x] = [y]"$, but there seems to be no further elaboration on why $x \sim y$ $⟺$ $[x] = [y]$ ...
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2answers
61 views

aRb if and only if a=b or a-b=2^n for a natural number, n

Is R reflexive? Is R symmetric? Is R transitive? I know a=b is an equivalence relation so it is reflexive, symmetric, and transitive. I know a-b=2^n is reflexive but not symmetric or transitive. Not ...
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1answer
23 views

Openess of sets given by equivalence relations in the quotient topology.

I'm trying to prove this: Let $R$ be an equivalence relation in $X$. Show that $A$ is open in $X/R .\iff \bigcup_{[x]\in A}[x]$ is open in $X$ One of the first things that come to my mind is ...
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2answers
28 views

Equivalence Relation Proof for modular arithmatic

Given this modular relation: $x^3 \equiv y \pmod{3}$ how would you go about proving the transitivity of the system? I have proven the reflexivity, and symmetry pretty easily but the transitivity is ...