This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric,...), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

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1answer
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My question is a very basic one about relations

I am learning about relations right now and I have a question about some terms. I am told a relation on $A$ is a subset of $A\times A$. Then I am told a relation $R$ on $A$ is reflexive if for all ...
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2answers
29 views

Number of Symmetric Relations on a set A

I'm having trouble understanding their explanation. I follow everything up to "The Set $A_2$ contains $(1/2)(n^2 - n)$ subsets..." could someone please help explain this to me? Source: Discrete and ...
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2answers
72 views

Relations $\rho $ and $\rho^2$ [closed]

If $\rho$ is a relation on a set $A$, define $\rho^2$ by $a\rho^2 b$ if and only if there exists $c$ with $a\rho c$ and $c\rho b$. If $\rho$ is reflexive/symmetric/transitive does $\rho^2$ have the ...
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1answer
133 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
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1answer
87 views

$A = \mathbb{R}$ , and $a\mathrel{p} b$ if and only if $\sin a = \sin b$

My question is: For the relation $p$ described below, determine if $p$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $p$ is an equivalence relation, describe the equivalence ...
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1answer
27 views

Transitive Relations on a set

I am trying to study binary relations (for myself, it's not an assignment!) I have the set $\{1,2,3,4\}$, and one of the relations in the exercise is $\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}$. A ...
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2answers
47 views

Can a relation be a partial order and an equivalence at the same time?

Can a relation be a partial order AND an equivalence at the same time? For instance, if we have a set A = {1, 2, 3, 4, 5} and a relation R on A defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}: ...
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1answer
39 views

How does one find/list equivalence classes?

Can someone explain how I would find/list the equivalence classes (And number of equivalence classes) of these two examples? Example 1: A is the set of all possible strings of 3 or 4 letters in ...
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1answer
42 views

How would I draw the diagram for this relation?

The question I am trying to solve is below. I have proven it is an order but am unsure how to draw the diagram for it. Can someone point me in the right direction? Let A = {1, 2, 3, 4}, and let R be ...
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1answer
30 views

How to prove this equivalence relation?

How would one go about proving this is an equivalence relation? I have no idea where to start. $\cal R$ is the relation on $\Bbb Z \times \Bbb Z$, such that $((a, b),(c, d)) \in \cal R$ if and only ...
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0answers
18 views

Special relations on a finite set

Given a set $S = \{s_1, \dots, s_n\}$, $S \times S$ is the product space of $S$ with itself. Let $S_0 = \{(s_i, s_i), i=1,\dots,n\}$. Are there a name and/or notation for the operation mapping $S$ to ...
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1answer
47 views

Find all equivalence classes

Let R by a relation defined on pairs $(m,n)$ of integers $m$ and natural numbers $n$ by $(i,j) R (k,l)$ if $il=jk$. Prove that this is an equivalence relation and give the equivalence cases. Show ...
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2answers
30 views

Transitivity of a relation [closed]

Is the relation {(1,2)(3,4)(5,6)} is a transitive relation. I have found in many references and ncert text that it is transitive. Give reason for u r answer.
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1answer
38 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
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1answer
20 views

Composition of relations: Incomplete proof.

Let $R$ be a relation from $A$ to $B$, and $S$ be a relation from $B$ to $C$, and $T$ be a relation from $C$ to $D$. I want to prove that $T\circ (S\circ R)=(T\circ S)\circ R$. This is how I proved ...
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1answer
48 views

Partial order relation (Antisymmetric property), given a relation $xRy \iff x-y\le 4$

Given the set: $A=\{1,2,3,\dots,19,20\}$. The relation $R$ is defined on $A$ as: $xRy\Leftrightarrow x-y\leq4$ Is $R$ a partial order relation? I know that for a relation to be partial order it has ...
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229 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
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2answers
34 views

Equivalence Relations (Discrete Math)

Hello I'm having trouble with this math problem on equivalence relations. Let X be any subset of the set of positive integers Z. Define a relation ~ on X as follows: I have reflexive proven, having ...
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1answer
17 views

Relation symmetric and antisymmetric

Let $A$ a non-empty set. If there is a complete relation on $A$ that is both symmetric and antisymmetric, does it imply that the relation is the "equality" and $A$ has one single element?
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2answers
19 views

Subset Relation: Is the subset relation a partial order?

I read in a Wikipedia entry (subset in german http://de.wikipedia.org/wiki/Teilmenge): "Every set is a subset of itself" But for example, if A is a set of all sets, with maximum 5 Elements, than A ...
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1answer
75 views

The composition of the $<$ relation with itself

I am struggle with answering this question. I do not understand how to approach this question. 1.Let <􏰈 denote the less than relation on the set of integers. Describe the squared relation <^2 ...
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1answer
51 views

Notation interpretation

Consider the set $$\Bbb R^n :=\{x=(x_1,...,x_n):x_1,...,x_n \in \Bbb R \}.$$ For $x,y\in \Bbb R^n$, we define $<$ as below: $$ x<y \iff \exists j \in \{1,..,n \} \left( x_j<y_j \wedge ...
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0answers
37 views

Implies the $\leq$ relation a lexicographical relation?

Consider the set $\Bbb R^n = \{ x = ( x_1, ..., x_n): x_1,...,x_n \in \Bbb R \}.$ For $x,y\in \Bbb R^n$, we define $ x<y \iff \exists j \in \{1,..,n \}(x_j<y_j)$ $\wedge \forall i \in \Bbb N ...
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2answers
69 views

Lexicographical order in $\Bbb R^n$?

Consider the set $ \Bbb R^n = \{ x = (x_1, ..., x_n) : x_1, ..., x_n \in \Bbb R \}.$ For $x,y\in \Bbb R^n$, we define $<$,$\leq$ as below: $$ x<y \iff j \in \{1,..,n \} (x_j<y_j) ...
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1answer
83 views

Lexicographical order

Hello everyone i'm trying to solve an exercise that contains the following istructions. Let it be $ \Bbb R^n = \{ x = (x_1, ..., x_n) : x_1, ..., x_n \in \Bbb R \}.$ Let define on $ \Bbb R^n$ a ...
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1answer
52 views

For semigroups, $S\preccurlyeq T$ iff there exists an injective relational morpism $\mu: S\to T$.

This is Exercise 1.16 of Howie's Fundamentals of Semigroup Theory. The Details. Definition 1: Let $A$ and $B$ be sets. A relation $\rho$ from $A$ to $B$ is a subset of $A\times B$. Define ...
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0answers
35 views

Is this relation reflexive, symmetric, antisymmetric, transitive, or whether it is equivalence relation

Is this relation reflexive? symmetric? transitive? Is it an equivalence relation? Explain. I know that for it to be an equivalance relation, it has to be reflexive, symmetric and transitive.
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1answer
23 views

Does (f(0)=g(0) or f(1)=g(1)) define a transitive relation on function?

I need is to check if a relation is an equivalence or not. I can see that it is reflexive and symmetric but I'm not able to find out if it is transitive. The relation is defined on the set of all ...
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1answer
24 views

Is every left-unique relation right-uniqe?

Lets say we have a relation A x B. As far as I unterstood, in a right-unique relation, for every element from A, there is at least one element in B. But there might be elements in B which do not have ...
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3answers
28 views

Why is this Relation R (graph) not transitive?

Let the arrow graph of R be the following: If we get the ordered pairs we have that R = { (a,a), (a,b), (a,c), (b,b), (b,a), (b,c), (c,a), (c,b), (d,d) } If we analyze this: *Reflexive - NOT ...
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1answer
40 views

Set of ordered pairs of the transitive closure R* of R

I pretty much know how to get the ordered pairs by doing the arrow graph method since the matrix method is much more complex. let R be: R = { (a,b), (b,a), (a,c), (c,d), (c,e), (e,c) } (I am ...
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2answers
32 views

How do you determine if a relation is transitive?

Suppose I have the relation P such that $$ x P y $$ iff $$ x = y^2 $$ How do I determine whether or not the relation is transitive?
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1answer
48 views

Venn diagram for a relation

My high school math book says the following diagram is a Venn diagram. But I think this is not correct. Is it right? If not, what is the following diagram that represents a relationship called?
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1answer
31 views

Steps to determine if a relation of a set is reflexive,symmetric or transitive?

I am having problem understanding these concepts. For example, let $A = \{2,3,4,5,6,7,8\}$. The definition I found says that $x R y \iff 3 | (x-y)$. How do I know if the relation $R$ on $A$ is ...
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1answer
38 views

Properties of a relation

$\cong\;=\{((x_1,y_1), (x_2,y_2))\in \mathbb R^2 ×\mathbb R^2 |x_1^2-x_2^2=3y_1^2-3y_2^2\}$ finitary relation meaning $(x_1,y_1) \cong (x_2,y_2)$ if $x_1^2-x_2^2=3y_1^2-3y_2^2$ Is this finitary ...
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1answer
15 views

equivalence relations proof over the same set

I want to proof the following theorem: Let R be an equivalence relation on set A. Then {R[a]:a that belongs to A} is a partition of A. So long I have manage to proof that each a that belongs to A, ...
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3answers
68 views

Recursive definition of the relation greater than on N X N

Give a recursive definition of the relation greater than on N X N using the successor operators s? I started this question throw this way: basis: (1,0) ∈ N x N could someone help me in recursive ...
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1answer
22 views

are these binary relations?

I have found the following examples of Binary Relations, but I am not pretty sure is the conclusion the author arrived is correct. X is a number of people x N y, implies that x lives next to y; for ...
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3answers
861 views

Symbol for unknown relation?

When solving equations like $$\begin{align} 4x-4 &=\frac{(2x)^2}{x} \\ -4 &= \frac{4x^2}{x} -4x \\ -4 &= 4x -4x \\[0.2em] -4 &= 0\end{align}$$ using the equality-symbol feels like ...
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1answer
23 views

Difference between Inclusion and continuation

Halmos defines the order continuation as follows: We shall say that a well ordered set A is a continuation of well ordered set B if B is a subset of A, if, in fact, B is an intial segment of A and ...
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1answer
20 views

$aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$ [Prove Equivalence Relation]

The question: $R$ is a relation on $\mathbb{N}$ defined by $aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$. Prove that $R$ is an Equivalence relation. The problem: I can define an ...
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2answers
38 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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1answer
23 views

Does an asymmetric relation entail an antisymmetric relation?

So if there exists an asymmetric relation within a set, does it also entail that there will be an antisymmetric relation in that same set? If so, then it is possible to find out whether a set ...
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1answer
30 views

How is a relation defined on ordered sets?

I am reading that $(\mathbb{Z}, \leq )$ is a total ordered set. I understand how it satisfies reflexivity, antisymmetry, transitivity. But it says that because for any $a,b \in \mathbb{Z}$, either ...
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0answers
12 views

Partial Order Matrix Representaion

What would be the general matrix representation of a partial ordering ? i.e. since the relation must be reflexive, mat(i,i) = 1.
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69 views

Good book for self-studying Binary Relations

I am studying economics and I frequently encounter Binary Relations. But without any good knowledge of it, I get confused. Here is some background, if it's helpful: I know calculus(single and ...
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1answer
53 views

Why is this relation recursive?

A relation $R \subset \mathbb{N}^d$ is called recursive if there exists a primitive recursive function f with $$ (x_1 ,\dots,x_d) \in R \Leftrightarrow f(x_1,\dots,x_d)=0.$$ In Kurt Gödel's article ...
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1answer
50 views

condition for transitivity

In transitive relations, $aRb$ and $bRc$ implies $aRc$. But what if there are no $bRc$, can we say that the relation is transitive? For example, are relations $R\subseteq V\times V$, corresponding to ...
2
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1answer
56 views

I have two symmetric relations on a set. How can I prove that the symmetric difference is irreflexive?

I have this problem. Let R and S be symmetric relations on a set A. Prove or disprove: $R \oplus S$ is irreflexive. Now I'm assuming it's not true, because $(x,x)$ can be an element of $R$ ...
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3answers
91 views

A big list of non-trivial examples of functions from outside mathematics

I will be teaching my students about functions, and want to stress that functions are not only the usual mathematical ones (linear, logs, exponential, ...), but that function is fundamentally a ...