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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process of $(a,b)R(c,d)\implies a\cdot b(b+c)=bc\cdot (a+d)$ being transistive relation..

My question was is as follows: If a relation $R$ defined as $\mathbb Z\backslash\{0\}\times\mathbb Z\backslash\{0\}$ as $(a,b),(c,d),(e,f) \in \mathbb Z\times \mathbb Z$ where $(a,b)R(c,d)\implies ...
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0answers
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drawing diagram for binary relation

im working on the practice problem on unit about sets and relations The question is: Let a = {1,2,3,4} and R be a binary relation on A x A given by: ((a,b),(c,d)) ∈R if and only if a divides c and b ...
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1answer
59 views

how to fnd if R is an order?

hello i have a upcoming quiz and I was solving practice problems that the instructor gave us. But Im not sure how to approach this problem the problem is: Let $A = \{1,2,3,4\}$, and $\mathcal{R}$ be ...
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1answer
78 views

How to show that R(binary relation on A x A) is an order?

im working on the practice problem on unit about sets and relations The question is: Let a = {1,2,3,4} and R be a binary relation on A x A given by: ((a,b),(c,d)) ∈R if and only if a divides c and b ...
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2answers
88 views

Relation that is only symmetric, reflexive, antisymmetric or transitive?

What could be a possible example of a relation that's symm, reflex, antisymm, transitive? I am working on practice problems on the unit about Sets and Relations. The question asks me to give a ...
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2answers
32 views

Anti-symmetric relation given by a matrix

Relation R is given by a matrix $$\begin{bmatrix} 1& 0& 0& 0\\ 1& 1& 0& 0 \\ 1& 0& 1& 0 \\ 1& 1& 1& 1 \end{bmatrix} $$ Is it anti-symmetric? I'm ...
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1answer
21 views

intersection of antisymetric relations is antisymetric

Suppose $A$ is some set, and $R$ and $S$ are relations on $A$ s.t. $R$ and $S$ are anti-symmetric. I want to prove that $R\cap S$ is anti-symmetric. Let $a,b \in A \ $ s.t. $a\ne b$ and $(a,b)\in ...
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1answer
28 views

Restriction of an equivalence relation on a subset.

If we have an equivalence relation defined on a set E and S its subset. Is the relation defined on S is also an equivalence one? Thank you for your answers.
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1answer
31 views

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
25 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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Is this relation an equivalence relation? If so, identify the equivalence classes. [on hold]

Determine if $ρ$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $ρ$ is an equivalence relation, describe the equivalence classes. $$A = \mathbb R \,\text{ and }\, aρ b \;\text{ ...
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2answers
71 views

Relations $\rho $ and $\rho^2$ [on hold]

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
81 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
83 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
23 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
33 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
27 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
33 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
29 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
17 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
34 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
27 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
21 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
17 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
39 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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1answer
182 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
29 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
13 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
17 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
51 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
50 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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35 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
59 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
66 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
47 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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29 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
21 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
16 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
22 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
18 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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30 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
42 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
27 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
37 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
44 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
21 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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124 views

two quantities P and Q exhibit a constant rate of change relationship

Given that two quantities P and Q exhibit a constant rate of change relationship, determine if each statement regarding these quantities is always true (A), sometimes true (S), or never true (N). ...
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824 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
17 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 ...