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

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3
votes
2answers
136 views

Proving if a relation is an equivalence relation

I have been able to figure out the the distinct equivalence classes. Now I am having difficulties proving the relation IS an equivalence relation. $F$ is the relation defined on $\Bbb Z$ as follows: ...
1
vote
0answers
75 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 ...
3
votes
1answer
52 views

Surjections and equivalence relations

(a) Let $f: A \to B$ be a surjective function. We define $a_1 \sim a_2$ if $f(a_1)=f(a_2)$. Prove that $\sim$ is an equivalence relation. Reflexivity: This comes for free. If $a_1 \sim a_1$, ...
3
votes
0answers
36 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 ...
0
votes
4answers
44 views

Bijection from set of equivalence classes to $\mathbb R$

In $\mathbb R[X]$, define an equivalence relation ~ by $P_1$~$P_2$ if $P_1-P_2$ is divisible by X. I have shown that $X$ is an equivalence relation. Let $\mathscr Q$ denote the set of equivalence ...
0
votes
1answer
62 views

Reflexive or Irreflexive

Are the following relations reflexive or irreflexive $R = \{ (x,y) : y = 2x\}$ $R = \{ (x,y) : x \text{ is a sibling of }y\}$ $R = \{ (x,y) : x = 3 + y\}$ I believe 1 is reflexive but I'm not sure ...
1
vote
3answers
88 views

Define a relation on the set of all real numbers $x,y \in \mathbb{R} $ as follows:

Define a relation on the set of all real numbers $x,y \in \Bbb{R} $ as follows: $x \sim y$ if and only if $x - y \in \Bbb{Z}$ Prove this is an equivalence relation and find the equivalence class of ...
0
votes
1answer
62 views

Equivalence relation $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$

Let $(G, \cdot)$ be a group with an Identity element $e$. (i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and ...
0
votes
1answer
31 views

For a given image $\mathbf X$, the equivalence class for pixels $p$ with labels $l$.

$\left[l\right]=\left\{p \in\mathbf X|\,p\sim l\right\}$. This is taken from this paper on image segmentation, page $2$. I don't know how to interpret this, do they mean "all the pixels on image ...
0
votes
0answers
53 views

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 ...
-2
votes
1answer
31 views

Formulate a relation $R$ between $2$ sets $A$ and $B$

Let $A$ and $B$ be $2$ sets of real numbers. How can I formulate the following entence, in mathematical terms, not plain english. IF At least one Element $x$ of $A$ is equal to one element $y$ of ...
1
vote
2answers
58 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 ...
1
vote
1answer
33 views

equivalence relation difficult problem

Let $R_1$, $R_2$ be 2 equivalence relations on $X$; prove that $R_1\cup R_2$ is an equivalence relation on $X$ if and only if $R_1\cup R_2=R_1\circ R_2$ I really don´t have any idea how to do it, I ...
0
votes
1answer
37 views

equivalence relation and quotient set problem

Let $R$ and $S$ be equivalence relations on X so that $X/R$=$X/S$, prove that $R=S$ how can I solve this problem?
2
votes
2answers
29 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...
2
votes
1answer
65 views

Understanding Pushouts in Top.

The Pushout of $X \leftarrow Z\rightarrow Y$ with $f:Z\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathbf{Top}$ exists and is given by $X\coprod Y/\sim,$ where "$\sim$ is the equivalence relation ...
-4
votes
1answer
61 views

Proving rational numbers [closed]

Definition 33: Define a relation $\sim$ on $\mathbb Z \times (\mathbb Z \setminus \{0\})$ by setting $(a,b) \sim (c,d)$ if $ad - bc = 0$. Proposition: The relation $\sim$ defined above is an ...
0
votes
1answer
15 views

How to prove this proposition that has to do with elements and equivalence relations

Every element $z$ in $X$ is in exactly one equivalence class. Not sure how to prove this. I proved that every element $z$ in $X$ is in some equivalence class by using the definition of $[x]$. How ...
0
votes
1answer
63 views

Prove this is an equivalence relation

$A$ is related to $B$ if $M_n(A)\simeq M_m(B)$ for some integers $m$ and $n$. Clearly reflexivity and symmetry are trivial. It's transitivity that I am struggling with. Is it the case that if ...
0
votes
4answers
53 views

Equivalence Relations

Review for Group Theory Final Exam: Define a relation on $\Bbb{R}^2 \setminus (0, 0)$ by letting $(x_1, y_1) \sim (x_2, y_2)$ if there exists a nonzero real number $\lambda$ such that $(x_1, y_1) = ...
0
votes
2answers
113 views

Reflexive, Symmetric, Anti-Symmetric relations

Let $A = \mathbb Z \times ( \mathbb Z\setminus {0} )$. A binary relation $R$ on $A$ is defined as follows: For all $(a,b),(c,d) \in A$ $$(a,b) \,R\,(c,d) \iff ad = bc$$ now how do I find if $R$ is ...
-1
votes
1answer
244 views

Quotient set definition

so, i search and search about quotient set and cant figure out what is this. At the beginning i think it was the same of partitions, but now i'm confuse. Can someone show some examples and explain? ...
0
votes
1answer
49 views

proving something is a well-defined function

1) Define ~ S4 as follows: for f, g element of S4 f~g if and only if f(4) = g(4) this is easily seen to be an equivalence relation on S4 (you don't have to show this) let X = S4/ ~ be the set of all ...
0
votes
1answer
57 views

Equinumerosity between equivalence classes set and power set

I´m currently working on the following problem: "Let $\xi$ = $\{ $ $\bot$ $\mid $$\bot$ is a equivalence relation over $\mathbb{N} $$\} $ Show that $\xi$ and $2^\mathbb{N} $ (power set) are ...
0
votes
2answers
77 views

Discrete Math - Equivalence Classes

I'm trying to understand a problem that my textbook gives me. Here is the problem: The relation $R$ is an equivalence relation on the set $A$. Find the distinct equivalence classes of $R$. $A = \{0, ...
1
vote
1answer
42 views

how many elements does Ia have?

Let $A=\{1,2,3,4\}$. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by $f,g \in F$ $f R g \Leftrightarrow |f(A)|=|g(A)|$ $f(A)=\{f(x): x\in A\}$ ...
1
vote
1answer
35 views

Need help to understand equivalence class

This is in my note Let S={1,2,3,4} Let R be the relation on P(s) defined by xRy <=>|x|=|y| how many equivalence classes are there ? 5 [∅]={∅} [{2}]={{1},{2},{3},{4}} [{2,3}]={{1,2},.......... ...
1
vote
0answers
25 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 ...
1
vote
1answer
87 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 ...
0
votes
2answers
56 views

Determine all equivalence classes of $xy>0$

Define the equivalence relation $R$ as follows: For $x,y\in\mathbb R$, $x$ is equivalent to $y$ if and only if $xy\geq 0$. Determine all of the equivalence classes of this equivalence relation.
-2
votes
1answer
51 views

Basic Equivalent Relations Question [duplicate]

For $x,y\in \mathbb R$ $x\sim y$ if and only if $x-y \in \mathbb Q$. I need help with the following questions: If $a \in \mathbb Q$, what is the equivalence class of $a$? If $a \in \mathbb Q$, prove ...
-2
votes
1answer
64 views

Cardinality of “x−y∈Q”-equivalence class of 1/2 √ [duplicate]

For x,y∈I:=[0,1] define the relation on I as x−y∈Q. How big (using cardinal number) is the cardinality of the equivalence class [1/√2]? I have tried to solve it by finding the equivalence class but ...
2
votes
1answer
19 views

What is the structure of a directed graph with vertex set A which has a relation R

I am studying for a test and found this question in the book: Let $R$ be an equivalence relation on the set A (Non-empty). Let $D_R$ be the directed graph with vertex set $A$ and an arc from $x$ to ...
0
votes
2answers
41 views

Multiplication on $\mathbb{Z}_6$

How can I find all solutions to $[2]x=[4]$ in $\mathbb{Z}_6$? For example, is $[2]\times 8=[16]=[4]$? is this a right way to solve?
0
votes
1answer
81 views

What are some concrete examples of kinds of relations in math?

I'm writing an undergrad philosophy paper. My take on the issue is that the conceptual problem I'm addressing is only a problem because the word 'is' and 'relation' are too slippery. By more precisely ...
0
votes
1answer
84 views

Equivalence relations on S given no relation?

On my assignment, one of the questions ask to list all equivalence relations on S and count how many are of partial orders. Let S = {u, v, w}. List all equivalence relations on S. How many of these ...
0
votes
2answers
25 views

Meaning of Quotient in this context

I was seeing the following problem a couple days ago: Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected? I ...
0
votes
0answers
34 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.
2
votes
2answers
52 views

cardinal of a quotient space

Suppose that we have an equivalence relation $R$ defined on an infinite set $X$ and that all equivalence classes are finite. Is it so that the cardinal of the quotient space of $R$ is that of $X$? If ...
0
votes
1answer
46 views

Equivalence relation of a group acting on a set

Let A be a set and G be any subgroup of S(A). G is a group of permutations of A. Assume that G is a finite group. If u∈A, the orbit of u is the set O(u)={g(u): g∈G}. Define a relation ~ on A by u~v ...
0
votes
1answer
35 views

Binary relations: transitivity and symmetry

I've been looking at some examples for transitivity and symmetry. Suppose $A=\{0,1,2 \} $ and the relation $R=\{ (0,0),(1,1),(2,2),(1,2),(2,1) \}$ Well for starters this is clearly reflixe since ...
0
votes
1answer
23 views

equivalence relation-showing that an operation is well-defined

Define $f: \mathbb{Z}_n \to \mathbb{Z}_n$ as $f([a]) = [a^2]$. Show that $f$ is a well-defined function. I am confused as to how I could show this.
0
votes
5answers
83 views

Number of equivalence relations with a fixed size

How can I find the number of equivalence relations R on a set of size 7 such that |R|=29? Any advice would be greatly appreciated! :D
0
votes
1answer
50 views

Binary relations, closures and equivalences

Let $R$ be the relation on $Z$ such that $xRy \iff x-y=c$. Well, what I have so far is $R=\{ 0,-1,1,0,-1,1,0 \cdots\}$ Is $R^* $ and equivalence relation? Why not? This is where problems start: I ...
0
votes
0answers
39 views

Suppose $R$ is an equivalence relation and define $[x]_R=\{y:\langle x,y\rangle \in R\}$

I am not sure if I have approached this the right way, or if my approach demonstrates an appropriate grasp of the question being asked. Any help would be most appreciated. (I also hope that I've ...
2
votes
1answer
19 views

Describing the equivalence classes of $X \sim Y \iff |X\setminus Y|<\infty \wedge |Y\setminus X|<\infty$

Let $\mathbb{N}$ be the set of all natural numbers and let $P(\mathbb{N})$ be the power set of $\mathbb{N}$. On the set $P(\mathbb{N})$ we define a relation $\sim$ by the formula: $$X\sim Y \iff ...
2
votes
1answer
29 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
0
votes
1answer
41 views

Reflexive, symmetric, and transitive relations

On $A = \left \{1, 2, 3, 4 \right \}$ $\left \{(1,1), (2,1), (1,2)\right \}$ is NOT reflexive because there's no $(2,2)$ in the set. It is symmetric. However, it is NOT transitive. I'm confused as ...
0
votes
1answer
161 views

Find the equivalence class of 0

R is a relation defined on the integers by $(a,b) \in R$ is $a^2-b^2$ and is divisible by 3. I set a or b to zero to get all the negative and positive values in the equivalence class. Although I want ...
0
votes
2answers
65 views

How to calculate equivalence relations

How can I calculate how many equivalence relations can be defined on a given set? For example: How many possible equivalence relations can be defined on S = {a,b,c,d}?