1
vote
1answer
21 views

Prove that if the relation (R) is symmetric and antisymmetric on the set X, there exists a Y subset of X such that R is the = relation on Y.

Prove that for any relation R on a set X that is both symmetric and antisymmetric, there is subset Y \subseteq X for which R is the relation = on Y. I will tell you what I know. I know that ...
0
votes
0answers
16 views

order definition [closed]

i'm in desperate need of your much appreciated expertise with trying to define the following order relation, guess it might be somewhat near a lexicographical order, not a straightforward alphabetical ...
1
vote
0answers
17 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 ...
0
votes
1answer
31 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
1
vote
2answers
30 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
1
vote
1answer
54 views

Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
0
votes
1answer
42 views

Determining Equivalence Relation on $\Bbb{Z}$

Alright, I have a homework problem which I have researched, read up on and I (think) solved. I just need someone to either confirm my answer (and re-affirm my knowledge) or explain why I am wrong. ...
1
vote
1answer
65 views

Determine if R is an equivalence relation

I've got this question: Consider the relationship $R$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d)$ (denoted by $(a, b) R (c, d)$) if and only if $ad = bc$. ...
0
votes
0answers
26 views

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 ...
2
votes
1answer
37 views

Prove transitivity of 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 ...
2
votes
1answer
37 views

$R$ is transitive if and only if $ R \circ R \subseteq R$

Question: Let $R$ be a relation on a set $S$. Prove the following. $R$ is transitive if and only if $ R \circ R \subseteq R$. Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a ...
0
votes
0answers
24 views

Equivalence Relations and Order? Homework help

Having some trouble on these questions: c) Describe a partial order on {1, 2, 3} that is not a total order d) Describe a binary relation on {1, 2, 3} that is both a partial order and an equivalence ...
2
votes
1answer
102 views

The closures of a binary relation

I have the following dilemma concerning the equivalence closure of a binary relation. Let $X$ be a nonempty set and $R\subseteq X\times X$ (i.e., a binary relation over $X$). Consider the following ...
2
votes
1answer
18 views

Checking the equivalence relations of sets

$S=\{0,1,2,3\}, R:SxS, (m,n)\in R \text{ if } m+n=4$. From the condition of $R$, I found that $R=\{(2,2),(1,3),(3,1)\}$. Now I have to see if $R$ is reflexive, symmetric, antisymmetric, and ...
0
votes
1answer
30 views

Equivalence relation example

On the Wikipedia page about Equivalence Relations, there is a simple example: Let the set $\{a,b,c\}$ have the equivalence relation $\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$. The following sets are ...
2
votes
2answers
537 views

How many equivalence relations on a set with 4 elements.

Let S be a set containing 4 elements (I choose {$a,b,c,d$}). How many possible equivalence relations are there? So I started by making a list of the possible relations: ...
0
votes
0answers
9 views

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 ...
0
votes
1answer
35 views

Proving that $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$

This assingment is preparation for exam. I need to prove with $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$ that $\sim$ is equivalence relatio. Can you tell me how to do this. ...
2
votes
1answer
22 views

Example of an equivlance relation that is transitive

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as $(m,n)\in$ R if and only ...
0
votes
2answers
38 views

Trying to understand an example of an equivlance relation that is symmetric

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as (m,n)$\in$ R if and only ...
0
votes
1answer
31 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
1
vote
1answer
24 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
0
votes
1answer
51 views

Shortcut method for proving equivalence relations

Define the relation R on N*N by: (x,y)R(z,w) if and only if x-z = w-y. Check whether R is an equivalence relation. Explain your answer My teacher answer is: Using the shortcut method: ...
0
votes
1answer
41 views

Equivalence class for a relation

Consider the equivalence relation on Z ! Z given by (m, n)R(p, q) if and only if mq = np: (a) Find the equivalence class represented by (2, 5). (b) Describe the set S of the equivalence classes ...
0
votes
1answer
40 views

Very Abstract Relation with points

So I have this question on relations, that I really cant understand. I mean, I cant understand the question to be honest. Suppose a set $X$ of points on the plane and we "stabilize" a point $O ∈ X$. ...
8
votes
3answers
154 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
0
votes
1answer
36 views

If $E_1$ and $E_2$ are equivalence relations, is $E_1\circ E_2$ an equivalence relation?

I'm given two equivalence relations $E_1$ and $E_2$ over a set A and need to show whether the composition $E_1 \circ E_2$ is reflexive, symmetric and transitive. I only managed to show that $E_1 ...
1
vote
1answer
80 views

Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...
1
vote
1answer
84 views

Prove or disprove question with equivalence relation, classes and quotient group

Let $A$ be a set and $R$ an equivalence relation on $A$. Prove or disprove: If $A$ is countable then all the equivalence classes of $R$ are countable. If $A$ isn't countable then the ...
3
votes
3answers
198 views

Equivalence relation question with functions

We'll define on the set: $A=\Bbb R^{[0,1]}$ the relation $R$ by $fRg$ if $f(0)=g(0)$. Make sure it's an equivilence relation. What is $[\cos x]$ ? Describe all the equivalence classes ...
1
vote
3answers
54 views

Proof of equivalence relation on a set

Let $X = \{(a,b) \mid a,b \in \Bbb Z; b \ne 0\}$. We define $(a,b)\mathrel R (c,d)$ iff $ad = bc$. Prove that $R$ is an equivalence relation on the set $X$. Which known set do the equivalence classes ...
2
votes
2answers
68 views

Properties of the relation $R=A\times B \cup B\times A$

A is a set. Let $B\subsetneq A$. $R=A\times B \cup B\times A$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, ...
1
vote
1answer
57 views

Properties of the relation $R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$

$A= \Bbb R \\ R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence ...
1
vote
3answers
87 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 ...
2
votes
3answers
117 views

Is this relation reflexive, symmetric and transitive?

Define a relation $R$ on the set of functions from $\mathbb{R}$ to $\mathbb{R}$ as follows: $(f, g) \in R $ if and only if $f(x) - g(x) \geq 0$ for all $x \in \mathbb{R}$ Is this relation ...
0
votes
3answers
26 views

Relation symmetric confusion

So Symmetric = (a,b), (b,a) Set = {<1, 1>, <1, 2>, <1, 4>, <2, 1>, <2, 2>, <3, 3>, <4,1 >, <4, 4>} I understand ...
1
vote
2answers
57 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 ...
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 ...
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
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},.......... ...
-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
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
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
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.
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 ...
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 ...
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 $x_1, \dots, x_n$. ...
1
vote
4answers
36 views

Stuck on equivalence relation question

I have been stuck on this question for a while. I was wondering for a set $A={1,2,3,4,5,6}$, given that its distinct equivalence classes are $\{1,4,5\},\{2,6\},\{3\}$, what is the equivalence relation ...
1
vote
2answers
58 views

Transitive, symmetric $R$ such that for all $x$, there is $y$ such that $xRy$, is an equivalence relation

I'm stuck at one particular task I'm working on. Here is the task: Let R be a transitive and symmetrical relation on $S$. Assume that for all $x \in S$ there is a $y \in S$ so that $xRy$. ...