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

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4
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Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where $R=\{(0,0),(0,4),(...
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37 views

Restriction to equivalence relation is equivalence relation

Let $\mathcal{R}$ be relation on $A$ and $A_0 \subseteq A$. The $\mathbf{restriction}$ of $\mathcal{R}$ to $A_0$ is defined to be the relation $\mathcal{R} \cap (A_0 \times A_0) $. $\mathbf{...
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42 views

Proving Equivalence Relation $xPy$ iff $ y = x + n\pi$ on The Reals

I am trying to prove that the relation P on $\mathbb{R}$ by the rule $\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$ From what I can ...
3
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160 views

Arrow kernel in category theory and generalized equivalence relation

let $F : \mathcal{C} \rightarrow \mathcal{D}$ be a functor from two categories. It looks like that there are various notions of kernels one could define for a functor. One could define the arrow ...
3
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87 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
3
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62 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let $\tau_X\subset\...
3
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48 views

Relationship between 2 sinusoidal signal data sets?

I'm trying to relate a near shore tidal signal (point A) to 3 points along a long model boundary (points B C D). I want to possibly have a relationship between B C D with which we can convert A ...
2
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31 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
2
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54 views

Partitioning functions into equivalence classes based on running time?

I'm studying for my midterm and doing some practice problems, and I would be grateful if someone showed how to solve this. From my understanding you have to partition the functions into equivalence ...
2
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117 views

Relative Identity vs Set Theory

The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity. In his own ...
2
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44 views

Prove $[(n, k)] = 7^{*} \iff n - k = 7$

Prove $[(n, k)] = 7^{*} \iff n - k = 7$ given $7^{*} = [(8,1)] \in \mathbb{Z}$ and $k < n \in \mathbb{N}$ using any facts about addition in the Peano System. $k < n \in \mathbb{N}$ is defined ...
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222 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) $...
2
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42 views

Equivalence of Two Different Irrational Numbers Converse

Equivalence of Two Different Irrational Numbers I came across this and follow the proof, but I'm wondering how you would prove the converse, that is, if 2 irrational numbers are equivalent then their ...
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14 views

Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding $2$ to both sides of an equation results in an equivalent ...
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18 views

Technical Language Usage: Verb for an Equivalence Relationship “Forgetting” an Attribute that is “Modded Away”

This question is one of English usage, but I'm sure only mathematicians can answer it for me. I want to say in a technical report that an attribute is "ignored" or "forgotten" by an equivalence ...
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43 views

How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
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32 views

prove that for any pair of natural, there is a power of 2 that separates the pair of natural

i need to prove that: $$ \forall i,j \in \{1, \_ ,N \} \subset \mathbb{N} \ \exists k \in \mathbb{N} / (r_{2^k}(i) \leq 2^{k-1} \wedge r_{2^k}(j) > 2^{k-1})\vee (r_{2^k}(i) > 2^{k-1} \...
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47 views

LEN-Model equivalency

Starting position is a principal-agent-model with incomplete information (moral hazard) and the following properties: Agent utility: $u(z)=-e^{(-r_az)}$ Principal utility: $B(z)=-e^{(-r_pz)}$ Effort ...
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21 views

equivalence class of kernel relation of floor function

By taking particular values of $k$, I found that equivalence class of $k$, $[k]=\{k,k+1,k+2,...,2k-1\}$, equivalence class of $2k$, $[2k]=\{2k,2k+1,2k+2,...,3k-1\}$ and so on, but how to present it ...
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10 views

Number of symmetric relations

Let set $A=\{1,2,3\}$ Find number of symmetric relations that can be defined on $A$ containing ordered pairs $(1,2)$ and $(2,1)$ is? Can someone give me some hint for this question?
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37 views

Prove that multiplication is well defined

Let $M = \mathbb{N} \ \mathbb{x} \ \mathbb{N}$. We define the following relation on $M$. Let $(a,b)R(a',b')$ iff $a + b'=a'+b$ We define the set of intergers $\mathbb{Z}$, to be the set of ...
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41 views

relations and equivalence classes

$S$ is the set of all equivalence relations on set $A = \{1,2,3,4,5,6,7,8,9\}$ in which one of the equivalence classes is $\{1,3,5,7,9\}$. What is the size of $S$? I don't even know how to start... ...
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23 views

Equivalence relation on $S^2$ and unit square make same space

I think I understand a bit of this task, but I hope someone can look critical to my answers: Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the ...
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36 views

Is there a relation name that only applies to equal vectors if they also have the same origin?

Vectors $\vec{A}$ and $\vec{B}$ are equal if they have the same magnitude and direction. Is there a mathematical relation defined to apply to them only if they also have the same origin? Actually, I ...
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79 views

Equivalence of Group Actions, Transitivity, and Conjugate Subgroups

Some Preliminary Definitions and Properties: Actions of a group $G$ on sets $X$ and $Y$ are equivalent if the corresponding action of $G$ on maps from $X$ to $Y$ fixes some bijection. In this case, ...
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85 views

Symmetric closure of the reflexive closure of the transitive closure of a relation

Give an example to show that when the symmetric closure of the reflexive closure of the transitive closure of a relation is formed, the result is not necessarily an equivalence relation. My attempt ...
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24 views

Problem with understanding natural number difference

Proofwiki says the following about difference in natural numbers: In the context of the natural numbers, the difference is defined as: $n−m=p⟺m+p=n$ from which it can be seen that the ...
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36 views

equivalence relations example

determine if the relation on $\mathbb{Z} \times \mathbb{Z}$ is an equivalence relation. $$R=\{((a,b),(c,d)) \in A \times A: a+3b=c+3d\}$$ What I have thus far I need to show that R is reflexive, ...
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28 views

Symmetry of a relation

Let $A\neq\emptyset$ and $\mathcal{R}\subset A\times A$ a binary relation on $A$ such that the domain $D(A):=\{x\in A\mid \exists y\in A \text{ such that }(x,y)\in\mathcal{R}\}=A$ and $\mathcal{R}\...
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27 views

Explicit Description for an Equivalence Relation

Given a set function $f : X \to X$ let $\sim$ be the equivalence relation $x \sim f(x)$. Contextually, I am working with the coequalizer of $f$ and $1_X$. I want to have as much information about the ...
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48 views

does the equivalence class of an element in a set is the set itself?

what does equivalence class mean? I am trying to understand I think that I am a little confused so let's take this example: Suppose that we have the relation $2x+3y$ is a number less than or equal ...
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22 views

Is there a term for irreflexive relations whose reflexive closures are equivalence relations?

Consider a relation $\sim$ which is irreflexive, that is: $$ \forall A \:A\nsim A $$ Further suppose the reflexive closure of $\sim$ is an equivalence relation. The reflexive closure of $\sim$, ...
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23 views

Defn of invariant SET under an equivalence relation?

This is, I am sure, an easy question but something I did not see in my undergrad years and wikipedia does not define it. For a function or a relation or a property to be invariant under an equivalence ...
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36 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
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78 views

The relationship between an equivalence relation, equivalence classes, and partitions?

So I'm struggling right now to understand the concepts behind equivalence relations, equivalence classes, and partitions. This is what I understand about all these topics right now: Equivalence ...
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71 views

How to construct a transitive relation?

Given a binary relation $\mathcal{R}$ on a set $A$, you can be assured that the relation $\mathcal{R} \cup \mathcal{R}^{-1}$ is symmetric. (I can give a proof of this, if you would like.) I wanted to ...
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68 views

Showing properties of equivalence relation

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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93 views

Equivalence Relations with equivalence classes

Consider the following relation on $\mathbb{Z} \times \mathbb{Z}$: $(x, y)\sim (x', y')$ iff $xy' = x'y$. (a) Prove that it is an equivalence relation. (b) Consider the set of equivalence classes $\...
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30 views

equivalence relation problem - checking

We define eqiuvalence relation $\equiv$ at set $P(\mathbb{Z})$ as follow: $A \equiv B \iff |A \setminus \mathbb{N}| = |B \setminus\mathbb{N}|$ a)find equivalence class for $\emptyset$ b)find ...
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62 views

Describe an equivalence class

On the set N x N, define the following relation: (a, b) ~ (c, d) if and only if a + d = b + c (a). Show that this is an equivalence relation I have shown that this is an equivalence relation by ...
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94 views

Finding Equivalence Classes for Infinite Sets

Let $R$ be the relation on the set of rational numbers $\Bbb Q$ defined as follows: for all $q, r \in \Bbb Q$, $qRr$ iff $q − r \in \Bbb Z$. Then $R$ is an equivalence relation on $\Bbb Q$. What is ...
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117 views

Equivalence relation over groups $a\asymp_sb :\rightarrow\exists n\in\Bbb Z:as^n=b$: terminology and decision problem

Let's define this relation over the elements of an infinite group $(G,\cdot,e)$ $$a\asymp_sb :\rightarrow\exists n\in\Bbb Z(as^n=b)$$ where $a^n$ is defined as follow 1)$a^0=e$ 2)$a^{n+1}=aa^n$ ...
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100 views

Is “to be married” a transitive relation?

If you define a relation on the set of people, given by $R=\{x,y : x\text{ is married with } y\}$. Is this relation transitive? I would say it depends: In the western culture: If $x$ is married with $...
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27 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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100 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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49 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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39 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 $\Bbb Q^+$...
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91 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 maximal ...
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85 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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227 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 ...