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

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7
votes
4answers
782 views

“$ x $ is a brother of $ y $.” Why is this not transitive?

I am working on a problem set at the moment, and while checking my answers I realized that I have listed "x is a brother of y" as a transitive relation, while the answers say that it is not. EDIT: I ...
0
votes
1answer
24 views

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$.

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$. I need to show the equivalence classes for $[0]_{\sim}$ and $[\sqrt{2}]_{\sim}$. Here is what I did: $[0]_{\sim}$ = ...
0
votes
0answers
4 views

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.
0
votes
1answer
27 views

Determining a relation if reflexive, symmetric, and transitive

I just get stuck in this relation and need to find if this relation is Reflexive/ Irreflexive or Neither, Symmetric/ Antisymmetric or Neither, Transitive or Not. $$W_1 = \{(a , b) \in \mathbb ...
0
votes
1answer
33 views

Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
2
votes
1answer
21 views

The equivalence relation generated by a relation

Let $X$ be a non-empty set and let $r\subseteq X\times X$ be a relation on $X$. Let $R$ be the intersection of all equivalence relations on $X$ that contain $r$. Prove that if $xRy$, then one of the ...
0
votes
1answer
84 views

Klein bottle and real projective two-space

I've been looking at equivalence relations on the unit square: $[0,1] \times [0,1]$ that give rise to various surfaces such as the m$\ddot{\mathrm{o}}$bius strip, but I'm not too sure about the Klein ...
-2
votes
1answer
25 views

RELATIONS and sets [on hold]

If given a set where $F = \{f : f(x) = ax^2 - abx, a,b \in \mathbb{R}\}$ and the relation is $\{\text{roots of A}\} = \{\text{roots of B}\}$. How can i prove that this is an equivalence relation? ...
0
votes
1answer
16 views

Make the set $R$ transitive

Let $R$ be a relation on a set $A=\{w,x,y,z\}$ defined by $R=\{(w,x),(y,x),(x,y),(z,z)\}$. Using the original relation, $R$, make the necessary minimal additions to make $R$ transitive. I thought ...
1
vote
0answers
32 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 $S^{'}$ ...
0
votes
3answers
25 views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
0
votes
1answer
31 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
3
votes
1answer
24 views

Describe the equivalence classes for each equivalence relation

Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left( x_1, y_1 \right)$ ~ $\left(x_2, y_2 \right)$ iff $y_1=y_2$. Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left(x_1, ...
0
votes
0answers
49 views

How does Dilworth’s Theorem apply to the set $\{0, 2, 6, 7\}$?

I'm having some serious problems with Dilworth's Theorem. My question is 'how does Dilworth’s Theorem apply to the set $\{0, 2, 6, 7\}$?'. Any help is appreciated.
0
votes
0answers
25 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 ...
2
votes
1answer
31 views

How to show the existence of a binary chain

Suppose A, B are finite binary chains that hold AB = BA (* is the concatenation operator). How can I show there exists a binary chain C such that A, B are of the form CCC...C?
2
votes
1answer
88 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
0
votes
1answer
21 views

Relations and Equivalence Sequences

A relation is defined on the set $A=\{a + b\sqrt{2} \; : \; a, b \in \mathbb{Q} \text{ and } a + b\sqrt{2} \neq 0\}$ by $xRy$ if $x/y$ is in $\mathbb{Q}$. Show that $R$ is an equivalence relation and ...
1
vote
0answers
23 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 ...
0
votes
1answer
20 views

Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
0
votes
0answers
21 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 ...
0
votes
1answer
21 views

Complementary Relation Proof

If $R$ is reflexive, prove that $R^c$ is irreflexive. If $R$ is asymmetric, prove that $R^c$ is reflexive. Where $R^c$ = complement of $R$. I just can't figure out anything to say for the first one ...
0
votes
1answer
30 views

Given the following, is there equivalence relation?

Let $n$ be an integer. On the set $F$ of all integer-valued functions of a set $A$, suppose we define $f$ and $g$ to be related if $f(a)\equiv g(a)\pmod{n}$ for every $a\in A$. Is this an equivalence ...
1
vote
2answers
108 views

Number of elements in an equivalence class

Let a set X = {1, 2, 3, 4, ... , 2015} and a set Y = {1, 2, 3, 4, ... , 271}. Let S be the relation on P(X) defined by: For all sets A, B, that are elements of P(X), (A,B) are elements of S if and ...
0
votes
1answer
26 views

Decimals and equivalence relations

I am told that decimals set up an equivalence relation on the Reals and that decimal numbers and the Reals are not the same thing. I believe this also clarifys the famous $.\bar{9}=1$. That $1$ and ...
0
votes
0answers
18 views

Equivalence Relations between integers, and proving

Can anyone help with this problem. A relationship R between element a and b on the set of integers is $a\equiv b \bmod{29}$ if and only if aRb. a) Prove that R is an equivalence relation b) ...
1
vote
1answer
42 views

Prove that S is an equivalence relation on P(X)

Let $X = \{ 1, 2, 3, \ldots , 2015\}$ and $Y = \{ 1, 2, 3,\ldots, 271 \}$. Let S be the relation on power set (X) defined by For all A, B in $\mathcal{P} (X)$, $(A, B) \in S$ if and only if $|A \cap ...
2
votes
1answer
57 views

Measure spaces s.t. $\mathcal{L}^1 = L^1$

I have two questions: 1, Give an example of a measure space such that $L^{1}(X,\mathcal{A},\mu) = \mathcal{L}^{1}(X,\mathcal{A},\mu)$. 2, State, and prove, a condition on $\mu$ which is equivalent ...
7
votes
4answers
223 views

How to find the appropriate equivalence class?

I would like find the system of representatives & equivalence class for $[(1,-1)]_{\equiv 1}$ and $[(1,-1)]_{\equiv 2}$ given: $R_1=_{def.} (x_1,y_1)\equiv(x_2,y_2) \Leftrightarrow ...
1
vote
4answers
55 views

Prove $R$ is an equivalence relation.

I think I'm on the right track. Set $S = N \times N$, and for any two members $(a,b),(c,d)$ of $S$, define $(a,b) \simeq (c,d)$ provided that $ad = bc$. Prove that $\simeq$ is an equivalence ...
-1
votes
1answer
41 views

cardinality of polynomial

What is the cardinality of the following sets? (Choose from finite, countably infinite, or uncountably infinite.) The set of polynomials of the form $ax+b$ with $a \in\Bbb N$ and $b \in\{0,1\}$ ...
1
vote
1answer
165 views

If S is finite and $x/\rho$ is an idempotent of $S/\rho$ then $x/ \rho$ contains an idempotent

Currently working on the following problem, need a little help with the solution to the last part, any hints? Q: Let S be a semigroup, and let ρ be a congruence on S. Prove that if e ∈ S is an ...
1
vote
1answer
186 views

What is the congruence class of $x^3\mod x^3+x+1$?

I have a given Polynom congruence with a Polynom $x^3+x+1$ ... so the set of the congruence classes is $\{0, 1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$ But what would look this like? $$x^3\mod x^3+x+1\equiv ...
3
votes
1answer
38 views

Find equivalence classes of x ~ y : <=> x-y ∈ Z

The equivalence relation is: $$X=\mathbb{R}$$ $$x∼y:⇔x−y∈\mathbb{Z}$$ I proved the relation properties but how can I find the equivalence classes? Also I was wondering whether the equivalence ...
2
votes
1answer
35 views

Equivalence classes, not sure how to approach

While I understand equivalence classes, I can't seem to grasp this problem. This is what I am working with: Let’s define a relation ∼ on $\Bbb R^2$ by $(x, y) ∼ (p, q)$ if and only if $(x, y) = (λp, ...
0
votes
1answer
20 views

Is this relation symmetric on $\mathbb{Z} \times ( \mathbb{Z} \backslash \{0\}) $

Define a relation $R$ on $\mathbb{Z} \times ( \mathbb{Z} \backslash \{0\})$ by $(a, b) R (x, y)$ iff $ay = bx$. Checking whether $R$ is symmetric. $(a,b)R(b,a) \implies a.a = b.b$ which is false ...
1
vote
1answer
27 views

Find equivalence classes (Solution with questions)

I have to find the relation properties and the equivalence classes. $$X = \mathbb{R}^{2}$$ $$(x,y) \sim (u,v) \Leftrightarrow x - y = u - > v$$ Showing the relation properties of the ...
0
votes
2answers
23 views

List the elements in the equivalence class $E_{(9,2)}$

Let $A = \mathbb{N} \times \mathbb{N}$ and define a relation $R$ on $A$ by $(a, b) R (c, d )$ iff $ab = cd$. Obviously $R$ is an equivalence relation. Problem: List the elements in the ...
1
vote
1answer
33 views

Modules in Morita Equivalence

In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3). How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
0
votes
1answer
15 views

explain transitive nature of the given set

A relation $R$ in the set of human beings in a town given by $$R = \{(x,y):x \text{ is wife of } y \}. $$ How is it transitive? Can you explain?
0
votes
2answers
61 views

Proofs on equivalence relations rational numbers

The relation R = {(x, y)|x − y is an integer} is an equivalence relation on the set of rational numbers. I'm kind of confused with this question and what it is asking me to do. In order to solve ...
-1
votes
2answers
55 views

Is this relation reflexive?

Suppose $R$ is a relation on $N_4=\{1,2,3,4\}$ such that $R\circ R = R$. How would I prove that $R$ is reflexive? Could you give me some tips about how to start off?
-1
votes
1answer
29 views

Relation Reflexive? [duplicate]

Suppose $R$ is a relation on $N_4=\{1,2,3,4\}$ such that $R\circ R=R$. How would I prove that $R$ is reflexive? I am geting this statement as false, Please Let me know , How to prove this ?
1
vote
0answers
61 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 ...
0
votes
1answer
16 views

Disc quotient that is homeomorphic to the pinched torus

I apologize for my previous post. There was a mistake. I want to write a quotient of the disc $D^2:={\{z\in\mathbb R^2;\parallel z\parallel \leq 1 }\}$ by an equivalence relation which is ...
1
vote
1answer
44 views

equivalences relation on set $A$

I ran into a Pure Math Contest Problem that was took 1 month ago on my Schools, and I do lots of search, but i couldent any progress to solve it. If $R_1$ and $R_2$ be a equivalences relation on ...
1
vote
2answers
24 views

Relation antisymmetry check

Hello the question I am having trouble with is Describe a binary relation on 1, 2, 3 that is reflexive and transitive, but not symmetric nor antisymmetric. I Have the answer ...
0
votes
2answers
33 views

Matrix Similarity Question

Show that similarity is an equivalence relation. More specifically, recall that we say $A, B \in M_{n \times n}(F)$ (set of $n\times n$ matrices) are similar if there exists an invertible $Q$ such ...
3
votes
1answer
46 views

Example of equivalence relation where reflexivity/symmetry is not trivial

For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some ...
-1
votes
1answer
47 views

How to find union and intersection of these relations?

Problem: Let $R_1$ and $R_2$ be the "divides" and "is the multiple of " relations on the set of all positive integers respectively. That is, $R_1 = \{(a,b) | a \text{ divides }b\}$ and $R_2 = \{(a,b) ...