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

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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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143 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 ...
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46 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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89 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The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity.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 ...
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40 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 ...
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52 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 ...
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210 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) ...
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41 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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13 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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19 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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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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64 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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46 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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56 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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27 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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25 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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60 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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114 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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82 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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52 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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40 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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35 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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74 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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83 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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87 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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35 views

What does a null relation on all real numbers look like?

I'm being asked to test for reflexivity, symmetry, transitivity, and antisymmetry on a null relation for all real numbers, i.e. $X=\mathbb{R}; R = \emptyset $. What would such a resulting set look ...
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36 views

Equivalence Set, Subsets

Just a quick question: If a = [A] and a belongs to N (set of all natural numbers) doesn't that mean that A is a subset of N? The reason I'm asking this is because I'm trying to prove the theorem ...
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46 views

The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
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29 views

Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
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Congruence induced by a subset.

Consider the additive monoid of natural numbers and calculate the congruence generated by $\{(2,3)\}$. I know the answer is that the congruence has 3 classes which are $\{0\}$, $\{1\}$, and the rest. ...
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29 views

How I can define an equivalence relation on the graph of a function?

Let $a∈ℝ$ being fixed. Let $g:ℝ→ℝ$ be a real analytic function with infinitely many zeros. I want to classify the points $c,d$ such that $g(a)=(a-u)g′(c)≠0$ and $g(a)=(a-w)g′(d)≠0$ where $c∈(u,a)$ and ...
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95 views

Finding correct equivalence matrix for all group representations

This question is linked to a previous one that I asked: Attempting to find a specific similarity (equivalence) matrix I have a group of 24 elements, with two generators. I need to find an ...
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42 views

Spherical Grid Identifcation

I'm trying to see how the lower half of these grids look like when I make the following identification onto the unit sphere: CLICK HERE TO SEE IMAGE In notation, how would one represent this ...
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60 views

Simple counting question related to equivalence classes

Let $S = \{1,2,3,...,10\}.$ Define the relation $\mathscr R$ on the power set $\mathscr P(S)$ of all subsets of $S$ by: for all $A,B \in \mathscr P(S),A\mathscr RB$ if and only if $N(A) = N(B)$. ...
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Prove that the following relation is an equivlance relation.

Prove that the following relation is an equivalence relation and determine how many equivalence classes R partitions the set $Z^{+}$ into. R = {$(a,b) | a∈Z^{+} ∧ b∈Z^{+} ∧ 10 | (a^{2}- b^{2})$} Any ...
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92 views

Function on equivalence relation on functions

Let $E$ be the set of all functions from a set $X$ into a set $Y$. Let $b \in X$ and let $R$ be the subset of $E \times E$ consisting of those pairs $(f,g)$ such that $f(b) = g(b)$. Prove that $R$ is ...
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71 views

wikipedia relation algebra example #3

Wikipedia article gives the following relation algebra example: 3 An important generalization of the previous example is the power set $2^E$ where E ⊆ X² is any equivalence relation on the set X. ...
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19 views

How to show the equivalence of an Euclidean norm to Euclidean distance?

I have the following problem: I have a set of $N$ vectors. I have some a priori information where the information for each vector comes from. If the two vectors $x_1$ and $x_2$ have been sampled ...
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50 views

Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
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Is bisimilarity an equivalence relation

I want to know if bisimilarity is an equivalence relation. I need to make a proof showing that this is true but I have searched and I can only find for branching bisimilarity.
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35 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
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27 views

Help Representing Equivalence Classes

In the set $\mathbb{Z}$, we define two integers $x$ and $y$ to be equivalent ($x ≈ y$) if and only if $x \operatorname{div} 10 = y \operatorname{div}10$. How would one select a representative from ...
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21 views

About finding a binary relation

Let $δ_{n},β_{n}$ two sequences of rational numbers. Assume that the points $$P_{p}=(δ_{p-1},β_{p-1})$$ $$Q_{p}=(δ_{p},β_{p})$$ $$R_{p}=(δ_{p+1},β_{p+1})$$ are colinear and assume also that the ...
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56 views

Why relation “parallel” on the set of lines in a plane not transitive?

My book says relation "parallel" on the set of lines in the plane not transitive. And the definition in the book given is : A relation $R$ on a set $A$ is transitive if whenever $aRb$ and $bRc$ ...
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61 views

Trying to prove Equivalency using Boolean Algebra

The question presented was to use boolean algebra to show that XY’Z + X’Y’Z’ + XY’Z’ + X’YZ’ ≡ XYZ’ + XY’Z + XY’Z’ + XYZ’ I've tried using various laws of Boolean algebra, but the answer that I ...
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22 views

Showing relation is transitive (a,b) $\in$ R iff 2|(a+b)

Let R be the relation on natural numbers defined by (a,b) $\in$ R iff 2|(a+b) show it is transitive.
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51 views

Help with logical equivalence proof regarding a lemma for equivalence relations.

So the question is this: Suppose A is a set. Let ~ be an equivalence relation on A and let a,b be elements of A. Then Ta = Tb if and only if a ~ b. I need to prove this statement to be true. I know ...
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37 views

Define another Equivalence Relation from Geometric Figure?

Define relation $W$ on $R^2$ by $(x_1,y_1)W(x_2,y_2)$ whenever $x_1−y_1=x_2−y_2$. I have the equivalence classes to be given by $f^{-1}(r)=\{(x,y)\mid x−y=r\}$, where $r∈R$. So how would you define ...