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

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3
votes
1answer
40 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
24 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
34 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
56 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
30 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
17 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
36 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
47 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) ...
0
votes
2answers
46 views

How to mathematically show that the relation is transitive?

Problem: Show that the relation $x R y$ iff $x \leq y$ is a poset over the set of integers $\mathbb{Z}$ My work: I know that to show the relation is a poset or a post order, I have to show the ...
5
votes
1answer
88 views

Question about right and left cosets.

I want to do a question about how my algebraic structures professor defined left and right cosets. I'll write here his way to present them. We first talked about quotient group. Let $G$ be a group, ...
0
votes
2answers
34 views

How to prove the relation is transitive?

Problem: Consider the relation R on $N$ defined by $x$R$y$ iff $2$ divides $x + y$. Prove that R is an equivalent relation My work: I know that to prove that a relation is an equivalent relation, ...
2
votes
2answers
50 views

Can a relation from A to some other set B also be considered symmetric?

Note: This definition is from Discrete Mathematics and Its Applications [7th ed, page 577]. This is my book's definition of a relation R on a set A My ...
0
votes
4answers
98 views

Is antisymmetric the same as reflexive?

Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598]. This is my book's definition for a reflexive relation This is my book's definition for a anti ...
0
votes
1answer
41 views

What is the equivalence class of this equivalence relation?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 26 pg 645]. Problem: What is the equivalence class of this equivalence relation? Relation {(0, 0), (0, 1), (0, 2), ...
1
vote
1answer
66 views

Equivalence Class.

Let R be the relation of congruence mod 4 on Z: a R b if a - b = 4k, for some k in Z. What integers are in the equivalence class of 31? How many distinct equivalence classes are there? What are ...
3
votes
0answers
139 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 ...
0
votes
1answer
22 views

Last step in proof of equivalence results in partition

Prove that: Given an equivalence relation $∼$ on a set $X$, the equivalence classes of $X$ form a partition of $X$. Well, I first define $O_x ={\{y:x∼y}\}$. If $O_x ∩ O_y \neq ∅$, so there is some ...
0
votes
0answers
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 ...
0
votes
2answers
33 views

Equivalence relation on a set of integers

I was wondering if the relation $X$ would be an equivalence relation only if the result is an even number. For example the relation $X$ is given by $a\ X\ b$ only if $a+b$ is even. Would this be ...
0
votes
1answer
27 views

Is transitive Relation closed under composition?

it's true that equivalence relations is closed under composition, i.e., if R is a equivalence relation RoR is so.(Because RoR =R) But this not imply that any transitive Relation is so. Briefly; i ...
0
votes
3answers
39 views

Equivalence Relation with dividing x and y integers

Define $x\sim y$ means 5 divides $(x - y)$ for $x$ and $y$ integers. Show that is an equivalence relation. Equivalence relation means it satisfies reflexity, symmetry, and transitivity. reflexive: ...
0
votes
1answer
32 views

Equivalence relations between ordered pairs of natural numbers

I have been looking into equivalence relations and trying to figure out if certain relationships would be considered an equivalence relation. Lets using the following relation X between ordered pairs ...
0
votes
2answers
47 views

If $R=K[X]/(X^n)$, can represent any element as polynomial with degree $<n$

Let $K$ be a field and $R=K[X]/(X^n)$ where $n \in \mathbb{Z}_{n\geq1}$ and $(X^n)$ is the ideal generated by $X^n$. We denote $x:=X+(X^n) \in R$, any equivalence class $r$ in $R$ has a representing ...
1
vote
1answer
35 views

Proving symmetry and transitivity

I want to prove $\mathbb{N} \sim \mathbb{Z}$ by indication of a bijection, thus the equipotency of the two sets. I know that I have to prove reflexivity, symmetry and transitivity. The reflexivity ...
0
votes
1answer
60 views

How to show a relation is/isn't reflexive, transitive, or symmetric

I was tasked with this: Define a relation on Z by setting x R y if xy is even. (a) Give a counterexample to show that R is not reflexive. How do I go about proving this? Do I express this ...
0
votes
1answer
21 views

Question regarding symmetric difference - Please check my work

Let $S$ be a relation on $P(\Bbb{R})$, such that $\displaystyle S=\{<A,B>\in\left(P(\Bbb{R})\right)^2.\left|A\triangle B\right|\le \aleph_0$ Is $S$ an equivalence relation? My try: ...
0
votes
1answer
36 views

How to define a relation

I was tasked with this question: Let X = {0,1,2,3,4}. Define a relation R on X such that x R y if x + y = 4. I don't understand/know what syntax I should use to define this as a relation. What ...
4
votes
2answers
122 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
0
votes
1answer
24 views

Relations: Transitivity, Symmetry, and Reflexivity

Each of the following subsets $R$ of the $(x, y)$-plane defines a relation on the set $\mathbb{R}$ of real numbers. Determine which of the axioms (transitivity, symmetry, reflexivity) are satisfied: ...
1
vote
0answers
39 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, ...
0
votes
1answer
62 views

Vector Spaces: difference between Equivalence Classes and Quotient Spaces

I'm reading Herstein's Topics in Algebra and Halmos's Finite Dimensional Vector Spaces. I think I understand that if $V_F$ is a vector space over $F$ and $W$ a subspace of $V$, then there is an ...
1
vote
0answers
53 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 ...
1
vote
1answer
37 views

Proof of a equivalence relation

A set $A$ is equipotent to a set $B$ $(A\sim B)$, if a bijection $f: A \rightarrow B$ exists. How to prove, that $\sim$ is a equivalence relation? EDIT: I understand the concept of reflexivity, ...
2
votes
3answers
61 views

Trouble understanding equivalence relations and equivalence classes

I've read up a bit on equivalence relations and equivalence classes, but I'm a bit unsure on the whole concept. From what I've read and equivalence relation, ~, between two mathematical objects $a$ ...
3
votes
4answers
383 views

When a determinant is zero

Is it true that if $C$ is a square matrix of size $n$ and $\det(C) = 0,$ then $C^n = O_n$ or the $0$ matrix? If yes, then why is that? I know that the reverse is obviously true, so I wondered if ...
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votes
1answer
10 views

For a preorder $R$ on a set $A$, show that $S=R \cap R^{-1}$ is an equivalence relation on $A$

Let $R$ be a preorder on a set $A$, and let $S$ be the intersection of $R$ and $R^{-1}$ (the relational inverse of $R$). Show that $S$ is an equivalence relation on $A$.
0
votes
1answer
22 views

Equivalence relation - Equilavence classes explanation

I have the following equivalence relation problem. $Let \ R\subseteq 2^S*2^S = \{(A,B):|A\cap T|=|B \cap T|\}\ where\ S=\{0,1,2\} \ and \ T=\{1,2\} \ Show \ that \ R \ is \ a \ equivalence \ ...
1
vote
2answers
709 views

Difference between Reflexive and Symmetric in Discrete Maths

Difference between Reflexive and Symmetric in Discrete Maths? This is what I understand: Reflexive -> <a,a=a>, <b,b=b> uses ...
0
votes
1answer
58 views

How to show the simple equivalence?

I come across the following equivalence of integrations: $$\int\left[-I \left(h\right){\partial h \over \partial \tau}+{\partial h \over \partial x}{\partial \over \partial \tau} \left({\partial h ...
0
votes
1answer
22 views

Determine if “$n \sim m$ iff $nm>0$” is an equivalence relation on $\Bbb Z$

Determine whether the given relation is an equivalence relation on the set. $n$ is related to $m$ in the set of integers if $nm>0$. So my teacher said this set is not an ...
0
votes
2answers
33 views

How many equivalence relations over $\mathcal P(\mathbb N)$ satisfy: $[\{8\}]_S=\{A\in \mathcal P(\mathbb N)|A\neq \{1\}\wedge A\neq \{2\}\}$

How many equivalence relations $S$ over $\mathcal P(\mathbb N)$ satisfy: $$[\{8\}]_S=\{A\in \mathcal P(\mathbb N)\mid A\neq \{1\}\wedge A\neq \{2\}\}$$ Just to make sure I understand, the ...