0
votes
1answer
49 views

Proving a relation is transitive

I am trying to understand transitive relations. I understand given that a set may have $\{(a,b)(b,c)\}$ it must contain $(a,c)$ for it to be transitive. But for longer sets I am getting confused in ...
1
vote
1answer
22 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 ...
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
42 views

$M_{R^n}$; how to derive $n$ for transitive closure?

When finding the transitive closure of a relation $R$, I convert $R$ into a boolean matrix $M_R$, and find the union between $M_R$ and its powers up to $n$. $$M_{R^*} = M_{R^1} \lor M_{R^2} \lor ...
0
votes
1answer
30 views

questions about a proof to a question (about relations)

1) $R,S,T$ are relations on the same set. Prove that $R(S\cup T)=RS\cup ST$ The proof that I stumbled upon was the following: $(a,b)\in R(S\cup T)⇒((a,x)\in R)∧((x,b)\in S∨(x,b)\in T)⇒(a,b)\in ...
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
2answers
30 views

Use of “for all” in definition of reflexive and symmetric relations.

My book says that a relation R on A is reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$ symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$ Although I ...
0
votes
2answers
52 views

understanding reflexive transitive closure

Suppose I have the following relation $$R = \{(1,1), (2,3), (3,1)\}$$ To make it reflexive we add the following missing pairs: $$ \{(2,2), (3,3)\}$$ Now I wonder how to find the reflexive transitive ...
0
votes
1answer
34 views

Help determining relations on the set {1, 2, 3}

I'm studying for a midterm and I just want to make sure that my understanding of these 2 problems that my teacher gave is logically sound. If you could take a look and give me some feedback I would ...
0
votes
2answers
37 views

If $A_1$ and $A_2$ are countably infinite, and that $A_1\cap A_2=\phi$. Then prove that $A_1 \cup A_2$ is countably infinite.

If $A_1$ and $A_2$ are countably infinite, and that $A_1\cap A_2=\phi$. Then prove that $A_1 \cup A_2$ is countably infinite. My work: As $A_1$ and $A_2$ are countably infinite, there exists a ...
2
votes
1answer
34 views

Problems with the definition of transitive relation

Recently I found this problem, which made me realize I have some problems with relations that are vacuously transitive. Problem: Assume that $R$ is a relation on $A$ and define the relation $S$ as ...
2
votes
1answer
26 views

Union of Cartesian squares

I am trying to find sufficient and necessary conditions for a relation to be representable as a union of Cartesian squares: $\bigcup_{i\in I}(X_i\times X_i)$ for some family $X_i$ of sets. One ...
1
vote
2answers
31 views

How do you call this feature and property of relation?

If an operator can be defined for $k$ operands, $k \in \mathbb N$, how do you call this feature of the operator? For example, "+" is such an operator. Similarly, for a relation $R$ on a set $X$, $R$ ...
0
votes
1answer
16 views

Are all relations that are symmetric and anti-symmetric a subset of the reflexive relation?

Simple question, but I can't seem to find a guaranteed answer. A symmetric set contains (a, b) if it contains (b, a), but an ...
0
votes
1answer
46 views

Help understanding a theorem on transitivity of a relation

The theorem states this: The relation R on a set A is transitive if and only if $R^n \subseteq R$ for n = 1, 2, 3,... What I'm reading is that the nth power of that set is transitive if the set ...
1
vote
1answer
45 views

Bijection on a component of a cartesian product

I have been recently studying relations and mappings and I have come across the following problem. Consider two non empty finite sets $I,J$ and their cartesian product $I\times J$. Let $f\colon ...
1
vote
4answers
58 views

I do not understand the definition of antisymmetric relations

OK, let A be a set and let R be a binary relation on A. In my class we say that R is antisymmetric if and only if for every a, b in A, if (a, b) in R and (b, a) in R then a = b. Fair enough, but what ...
1
vote
1answer
33 views

prove a set relation R is transitive?

I have been thinking this problems all the evening, please help Let R be a relation on A. Prove that if Dom(R)∩ Range(R) = Ø, then R is transitive. Oh my god, how to prove this???
2
votes
2answers
545 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
18 views

Prove number of Relations on A with m elements and B with n elements

How would you prove that a set A with m elements and a set B with n elements has 2^mn relations?
0
votes
1answer
41 views

Reflexive relation on set of $n$ elements

How many reflexive relations are there on a set of $n$ elements? I did the problem and I got the answer $2 ^ {n ^ 2}$. Is it correct? Thanks for the help..!!
0
votes
1answer
20 views

Partial order up to equivalence

In certain contexts one runs into something like a partial order, but the antisymmetry property is weakened as follows: if $x \preceq y$ and $y \preceq x$ then $x \simeq y$, where $\simeq$ is a given ...
1
vote
3answers
62 views

How to write this set?

I hope someone can help me out here :) We have to sets : STUDENTS » All the students of the school CLASSES » All the classes of the school And the relation : STUDENTSCLASSES » Relates the students to ...
0
votes
1answer
20 views

Set Theory - Given 2 sets, are they order-isomorphic

We are given the sets $A=(1,2]\cup ((3,4)\cap \mathbb Q)$ and $B=(1,2)\cup ((3,4)\cap \mathbb Q)$ with the standard order $\leq$ of the reals. Are they order-isomorphic? Meaning, is there a bijective ...
0
votes
1answer
24 views

Simple set theory - Show a set is finite

let $(X, \leq_*)$ be a partially ordered set. Assume there is an isomorphism $f: (X,\leq_*) \to (\mathbb Z, \leq)$ let $A \subseteq X$ be a well ordered subset of $X$ with an upper bound. Meaning ...
0
votes
1answer
151 views

Count number of binary relations between sets

He, I have following questions: We have sets $A$ and $B$, $\left | A \right | = m,\left | B \right | = n$. 1) How many binary relations are there from $A$ to $B$? 2) How many binary relations are ...
5
votes
1answer
57 views

Question about sets of well-orders and isomorphisms between well-orders

I was thinking about this problem in trying to better get a feel for and understand well-orders (I'm trying to learn some set theory) - right now, I'm not really hoping for anybody to supply me with a ...
3
votes
1answer
51 views

Can a relation be transitive when it is not reflexive?

Lets say I have the following set: $$ \{1, 2\}$$ and on it the following relation is given: $$\{(1, 2), (2, 1)\}.$$ Now is the above relation transitive? My confusion: we can see, that it is ...
0
votes
3answers
50 views

Class Transitivity Proof

Prove that a class $T$ is transitive iff $\bigcup_{t \in T},\, t \subseteq T$ iff $a \in t$ whenever $a \in b$ and $b \in T$. I know that I need to begin by proving the first statement implies ...
1
vote
1answer
38 views

proving antisymmetry of partition refinement

Suppose $P$ is the set of all partitions of some set $S$. $R$ is a binary relation on $P$, the refinement relation, defined as $(\Pi_1,\Pi_2) \in R $ if and only if for every $S_1 \in \Pi_1$, there ...
0
votes
1answer
46 views

Determining why this is transitive

Why $R_3 = \lbrace (1,2),(3,4)\rbrace$ is transitive? It's like, transitive is said because there's $\{a,b\}$,$\{b,c\}$ then there will be $\{a,c\}$ right? But then, why is that one is said to be ...
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 ...
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
85 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 ...
0
votes
1answer
13 views

A criterion for complete lattice.

Is there an infinite partially ordered set $(X,\le)$, in which for each $A\subseteq X$, either $\inf A$ or $\sup A$ exists but for some $A\subseteq X$ either $\inf A$ or $\sup A$ does not exist.
1
vote
1answer
49 views

Quotient group with functions and relations question

For the set $\mathbb Z/5 \mathbb Z $ (the quotient group of $\mathbb Z$ with the relation R that is defined by $xRy$ if $5|y-x$) We'll define the following operations (both are $\cdot, +$ ...
3
votes
3answers
199 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 ...
1
vote
2answers
50 views

Function and Relations

I have a small question, if anyone could shed some light I would be really grateful! we have this relation R: ∀ x,y ∈ "≈" , [(x≈y) ⇔ (| x - y| ≤ 0.5)] Also relation R belongs to real numbers ℝ. ...
1
vote
1answer
107 views

How to prove relation is asymmetric if it is both anti-symmetric and irreflexive

Prove a relation is asymmetric if it is both anti-symmetric and irreflexive (anti-reflexsive). I tried to go from the definitions of the relations: Anti symmetric: $\forall x,y \, (xRy \land ...
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
1answer
50 views

Proof that there is an order relation

For an arbitrary set M there is a relation $R \subseteq 2^M \times 2^M$ about $$ A \mathrel R B \Leftrightarrow A \cup \{x\} = B$$ The join is a disjoint join. There are not more details what is $x$. ...
1
vote
3answers
35 views

Help understanding what's required for a relation to be symmetric?

Let $\,X=\{1,2,3,4,5\},\;$ does a symmetric relation on $\,X\,$ need to have all the elements of $\,X\,$ in the relation? Or can if have just a few elements of $X$ like this relation: $\,A ...
1
vote
1answer
51 views

I want to show that $\mathrel{R_1}$ and $\mathrel{R_2}$ are a partial order on $\mathbb N$, how would I do this?

Let $\mathrel{R_1}$ and $\mathrel{R_2}$ be relations on $\mathbb N$ defined by $x\mathrel{R_1}y$ if and only if $y = a + x$ for some $a \in\mathbb N_0$. $x\mathrel{R_2}y$ if and only if ...
1
vote
1answer
50 views

Proving that $\bigcup_{n=1}^{\infty }R^n$ is a transitive relation on A

Let $R$ be a relation on set $A$. How can i prove that $\bigcup_{n=1}^{\infty }R^n$ is a transitive relation on $A$? Maybe it has to do with $\bigcup_{n=1}^{\infty }R^n=tc(R)$ ?
-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 ...
5
votes
1answer
54 views

Why is this binary-relation antisymmetric?

Definition of antisymmetric binary-relation is $$\forall a,b\in\mathrm{A},\left[ \left(aRb\wedge bRa\right)\rightarrow\left(a=b\right)\right].$$ Let $\mathrm{A}=\left\{a\mid ...
0
votes
1answer
73 views

Maximal and minimal elements of the partial order relation

Let $A=\{1,2,3,4\}$ and $H$ is the set of antisymmetric relations on $A$. I think that $H = \{(1,2),(2,3),(3,4),(1,3),(1,4),(2,4)\}$. How would I find the minimum/maximum and min/max elements?
1
vote
4answers
139 views

is the empty set a relation?

Is the empty set is a relation? I wonder if the empty set is a relation.in enderton's a relation is a set of ordered pairs. If yes it's a relation why is that?. There is an example in the text for a ...