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

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For an integer n ∈ N define P(n) = {primes p : p is a factor of n}. [duplicate]

In other words, P(n) is the set of primes dividing n. For example P(12) = {2, 3} and P(1) = ∅. (a) Write down P(30) and P(32). (b) Consider the relation R on Z defined by n R m if and only if P(n) = ...
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0answers
15 views

prove the following equivalence $\ln(-\frac{l-x+\sqrt{r^2+(l-x)}}{l+x-\sqrt{r^2+(l+x)}})={}$ArcSinh$(\frac{l+x}{r})+{}$ArcSinh$(\frac{l-x}{r})$

Hi guys i'm trying to prove the following equivalence but i'm having some problems: $$\ln\left(-\frac{\ell-x+\sqrt{r^2+(\ell-x)^{2}}}{\ell+x-\sqrt{r^2+(\ell+x)^{2}}}\right) = \operatorname{ArcSinh} ...
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2answers
74 views

Equivalence class for the relation of having the same set of prime divisors

For an integer $n\in \mathbb{N}$ define $P(n) = \{p : p \mid n \text{, where $p$ is a prime} \}$. For example $P(12)=\{2,3\} $ and $P(1)=\emptyset$. Question: Consider the relation $R$ on ...
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How would I show that R is an equivalence relation?

If I were to consider the relation R on ℤ defined by n R m if and only if P(n)=P(m). How would I show that R is an equivalence relation? Any help is appreciated.
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1answer
61 views

Is the relation $a\mathrel R b \iff f(a) \equiv f(b)$ an equivalence relation?

Suppose that I have a relation $R$ of the form $a\mathrel R b \iff f(a) \equiv f(b)$, where $\equiv$ is an equivalence relation. In general, is $R$ also an equivalence relation? If not, what are the ...
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1answer
15 views

Comparing two circularly shifted matrices

I am looking for a way to compare two matrices A and B where B is the result of circularly shifting rows of A i.e. A = [1 2 3;4 5 6], B = [4 5 6;1 2 3] Is there an operator or metric that would ...
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1answer
30 views

Let R be the relation on ℤ+→ℤ+ defined by (a,b)R(c,d) if and only if a-2d=c-2b. List all the elements of the equivalence class [(3,3)].

I'm confused on how to find all the elements. I know how to find some but not all, wouldn't they be infinite? This is affecting me with the other questions as well. Thanks in advance!
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1answer
25 views

Geometric/visual interpretation of transitivity for equivalence relations on $\mathbb{R}$

If we graph equivalence relations on $\mathbb{R}$ on the plane $\mathbb{R} \times \mathbb{R}$, the properties of reflexivity and symmetry give rise to certain geometric properties--i.e. reflexivity ...
2
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2answers
33 views

Is $R=\left \{ (a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d) \right \}$ an equivalence relation on $X$?

Let $X= \left \{ a,b,c,d \right \}$ and $R=\left \{ (a,a),(a,b),(b,a),((b,b),(c,c),(c,d),(d,c),(d,d) \right \}$. I want to show that $R$ is an equivalence relation on $X$. My work: $R$ is reflexive: ...
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1answer
24 views

suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$

Suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$. Show that $$A=\bigcup_{i=1}^k A_i$$ Since $A_i\subset ...
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1answer
48 views

Expressing an formula in term of another one

I have this formula $$-\frac 1\lambda\left[\lambda D+1+W_{-1}\left(-r\exp(-\lambda D-1)\right)\right]$$ with $r$ , $\lambda$ and $D$ >0. Where $W$ is the Lambert W function ...
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0answers
37 views

Solve for ? - undetermined inequality symbol

So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example ...
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2answers
122 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
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1answer
16 views

Proving Equivalence Relations and Quotient Sets

Prove that the relation ∼ on $Z×Z$ given by $(a, b) ∼ (c, d)$ if $a+d = b+c$ is an equivalence relation. Give the quotient set $Z × Z/$ ∼.
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46 views

Geometric meaning of reflexive and symmetric relations

A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of ...
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2answers
56 views

How many times can transitivity property be applied

Can transitivity property be applied for infinite number of times for a certain problem??
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0answers
24 views

Axioms of equivalence relation in terms of the subset $R$

..An equivalence relation on $S$ is determined by the subset $R$ of $ S \times S$ consisting of those pairs $\left(a,b\right)$ such that $a \sim b$. Write the axioms for an equivalence relation in ...
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37 views

Properties of relations on Z

$$R_3 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|\frac{x-y}{5} \in \mathbb{Z}\} $$ $$R_4 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|x > y \} $$ I need to know which one is reflexive, symmetric, ...
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1answer
41 views

(solved) equivalance relation property (homework)

I am taking a first class in real analysis and am stumped already. The text leaves out the steps to the following relationship and I can not connect the dots. Any explanations would be appreciated. ...
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3answers
76 views

What is it called when !(a < b) and !(b < a) implies a = b?

I thought it would be some kind of symmetric equality but its impossible to do a google search on this, all I get are definitions of reflexive, symmetric and transitive. I'm not really sure which ...
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1answer
31 views

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
50 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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2answers
57 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
49 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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0answers
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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0answers
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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3answers
28 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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3answers
94 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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1answer
23 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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1answer
18 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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3answers
45 views

Giving an equivalence relation that corresponds to set partitions

My question is: Give equivalence relation that corresponds to the partitions A1 = {1,3,5} A2 = {2} A3 = {4,6} of the set A = {1,2,3,4,5,6} I don't know what the format of the relation should be, in ...
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1answer
27 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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3answers
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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1answer
50 views

An equivalence relation $\rho$ on $\mathbb R^2$

Define an equivalence relation $\rho$ on $\mathbb R^2$ by $(x_1,y_1)\rho(x_2,y_2)$ iff $x_1^2+y_1^2=x_2^2+y_2^2$ Then find the corresponding quotient space $\mathbb R^2/ \rho.$
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1answer
73 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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2answers
33 views

Set difference is finite - transitive relation?

Let $A=P(\mathbb N)$. The relation $E$ is defined: $(X,Y) \in E$ iff $X \setminus Y$ and $Y \setminus X$ are finite. I was given to prove this is an equivalence relation, however I had troubles ...
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2answers
46 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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1answer
16 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
34 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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2answers
39 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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0answers
29 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
34 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
48 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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2answers
51 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
37 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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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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29 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
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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1answer
60 views

Listing equivalence relations class for $x\sim y \Leftrightarrow x^2=y^2$ [duplicate]

Can someone help me on the right track for my proof for the statement below. I started and got stuck but I need help. Please guide me to answer what this statement requires and how to word it out ...
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0answers
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 ...