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

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Colored Picture for Equivalence Classes, Relations, Partitions, ..

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
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41 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 ...
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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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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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193 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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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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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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Finding Equivalence Classes for Infinite Sets

Let R be the relation on the set of rational numbers Q defined as follows: for all q, r ∈ Q, qRr iff q − r ∈ Z, where Z is the set of integers. R is an equivalence relation on Q. What is the ...
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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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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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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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42 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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38 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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32 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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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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78 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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118 views

Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,B⊆P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim $ is an equivalence relation.
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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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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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33 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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36 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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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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85 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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40 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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53 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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85 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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70 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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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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Define f:Z/3Z→Z/3Z by f([a])=[2a+1]

Just finished proving this to be injective, and well-defined. How would you prove it to be surjective? I understand surjective means that every element in the codomain is being used, and thus is the ...
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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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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 ...
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Poker Hand Equivalent Relation

Let $P$ be the set of all possible poker hands. Define a relation $J$ of $P$ by $a$ is $J$-related to $b$ iff $a$ and $b$ have no cards in common. Is $J$ reflexive? Symmetric? Transitive? Having a ...
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Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$

Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$ $cl_{eq}(R) = \bigcap\{S | $ S is an equivalence relation and $R \subseteq S\}$ is the equivalence closure of R. $R^* = \bigcap\{S | $ S is reflexive, ...
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Well-defined functions using mod $p$ equivalence classes

Prove that if $m, n$ are elements in the set $\mathbb{Z}$ and $m \equiv n \pmod p$, then $m^2 \equiv n^2 \pmod p$. Also, is the function $f: [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = ...
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What is the intersection of thses equivalence relations?

Let $S$ be the following subset of the plane: $$ S \colon= \{ \ (x,y) \ | \ y=x+1, \ 0 < x < 2 \ \}.$$ Then how to describe the equivalence relation $T$ on the real line that is the ...
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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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Transitive, Reflexive, Symmetric

So i know what each of these properties are..but this question does not provide any information on a 3rd variable so i was wondering how i would do it? ...
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29 views

A proper term for equivalence relation with a finite quotient set

Is there a proper term for an equivalence relation $\sim$ on some set $M$ such that it partitions $M$ into finitely many equivalence classes? Finite equivalence relation? or co-finite? or equivalence ...
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156 views

About Kernel and the coimage of a function

Introduction I was serching for a concept of "equivalence relations" induced by an arbitrary function in a "natural" way and I found the concept of Kernel. But I'm not sure that I understand it and ...
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Prove congruence relation.

Let $\sim$ be a relation where $x \sim y \Leftrightarrow x = y \lor (x \in I \land y \in I)$. Show $\sim$ is a congruence relation on $S$ where $S= \{a,b, c, d, e\}$ and $I = \{a, d\}$. One of the ...
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Question about Equivalence relation and partition

This is my first time of Abstract algebra, and I don't know how to solve this problem. Although I have an idea to solve this problem, I can't assure whather it is correct or not. Please show me how ...
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Bijection between partial order and $<$

How to show the correspondence between a less than relation and partial orders ? here A less than relation $<$ on a set $S$ is a relation that satisfies If $a < b$ , then $a \neq b$. ...
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Prove that $R_1$ refines $R_2$ if and only if the partition with respect to $R_1$ is a refinement of the partition with respect to $R_2$.

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Prove that $R_1$ refines $R_2$ if and only if the partition with respect to $R_1$ is a refinement of the partition with respect to $R_2$. ...
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39 views

Is an equivalence class same as residue class?

So my teacher's notes say this: An equivalence class consists of those integers which have the same remainder on division by n. They are also known as "congruence classes modulo n" Since ...
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What kind of relation does an ambiguous organization scheme introduce?

Please consider an arbitrary set of items, i.e. products in a supermarket or news in a news channel. Let's say we want to apply an organization scheme to that items. There are unambiguous ones, like ...
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A question on Partitioning regarding equivalence relations

Let $S$ be the Cartesian coordinate place $\mathbb R \times\mathbb R$ and define the equivalence relation $R$ on $S$ by $(a,b) R (c,d)$ iff $b-3a = d-3c$ Find the partition $D$ determined by $R$ by ...
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What is this equivalence relation explicitly?

Let $S \colon = \{ \ (x,y) \in \mathbf{R}^2 \ | \ \ y = x +1, \ \ 0 < x < 1 \ \}$, and let $T$ be the intersection of all the equivalence relation on the plane that contain $S$. Then how ...