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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Is there a standard name for the relation $X \times Y$?

Given sets $X$ and $Y$, is there a standard name for the relation $f : X \rightarrow Y$ whose graph equals $X \times Y$? Something like the full/maximum/top/total/complete relation?
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107 views

Directed and projective limit in Rel

I'm looking for the directed (or equivalently projective) limit of a directed family of relations in the category of sets with relations between them. Consider a family $\{ R_{ij} \subseteq U_i ...
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69 views

Where can I learn more about the concept that is dual to “relation”?

Let $X$ and $Y$ denote sets. Then a relation $X \rightarrow Y$ is, by definition, a subset of $X \times Y$. Dually, we can define that a "corelation" $X \rightarrow Y$ is a partitioning of $X \uplus ...
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Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
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43 views

Why is this not a poset after adding zero?

The problem    Consider the following set for divisibility. {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 96}. If 0 is added, the divisibility relation set will no longer be a poset. Please ...
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39 views

Divisibilty as relation set on $(\mathbb N \setminus \{0,1\})$

So i have to see if $\prec$ is order relation where two elements $(a,b)$ and $(c,d)$ are in relation $\prec$ if $a|c$ and $2b^{2}+6b\leq2d^{2} + 6 d$. This relation is defined on set $(\mathbb N ...
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56 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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37 views

Which of the following is always true for A and B

Given that: $ P(A) = 0.5$ $P(B) = 0.7$ $P(A \cap B) = 0.3$ I have to choose one option that is true... However they all seem to be false which means I am possibly making a mistake.. The only option ...
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484 views

Proving Reflexivity, Symmetry and Transitivity of a Relation

I'm currently taking an intro discrete math course, and I'm having some trouble understanding the rules of reflexivity, symmetry, and transitivity. The book isn't making a lot of sense to me, and my ...
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40 views

Prove $[(n, k)] = 7^{*} \iff n - k = 7$

Prove $[(n, k)] = 7^{*} \iff n - k = 7$ given $7^{*} = [(8,1)] \in \mathbb{Z}$ and $k < n \in \mathbb{N}$ using any facts about addition in the Peano System. $k < n \in \mathbb{N}$ is ...
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77 views

Using the ELO Rating System on Static Objects

The Setup Suppose we have a list of movies $m_1, m_2, \dots, m_n$ that we wish to rank in order of "quality." We define the "strength" of a movie $a$ by a function $f$ which takes in numerical ...
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24 views

How would I show the relations on this set of S?

I want to show that the set below is reflexive, anti-symmetric, and transitive. Let $S$ be the set of positive integer divisors of $180$ and consider the relation $\mid$ on $S$. I understand that ...
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297 views

Suppose $R$ is a relation on $A$ and $R^{-1}$ is the inverse. Proof or counterexample

Suppose $R$ is a relation on $A$ and $R^{-1}$ is the inverse. Give a proof or counterexample for each of the following statements. (a) If $R$ is reflexive, then $R^{-1}$ is reflexive. Let $x \in A$; ...
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95 views

How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
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43 views

Terminology for some special sets

The following predicates are used in the definition of total boundness of uniform spaces: a space is bounded if the predicate holds for every entourage, specifically for the first predicate (the ...
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790 views

Equivalence of norms is a equivalence relation

Two norms $||-||_1 $, $||-||_2$are equivalent if: for two constants $a,b$ and $x$ from $V$ a vector space over a field it holds that : $$a||x||_1\leqslant ||x||_2\leqslant b||x||_1.$$ This is a ...
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131 views

positive definite binary matrix

What are the conditions for a binary matrix $A$ (matrix with elements 0 or 1) to be positive definite (not even symmetric), i.e. $\forall x\neq 0, x^TAx>0, A_{ij}\in \{0,1 \}$ Put it another way, ...
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268 views

Examples of proofs by induction with respect to relations that are not strict total orders.

I have read this Wikipedia article and found it fascinating. I came across it after I tried to prove a certain statement with a method resembling induction in the set of natural numbers but ordered by ...
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146 views

How to describe all continuous maps from T to T'?

How to describe all continuous maps from $T'$ to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and $T'=\mathbb{R}$ with the topology with basis given ...
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61 views

How to construct a transitive relation?

Given a binary relation $\mathcal{R}$ on a set $A$, you can be assured that the relation $\mathcal{R} \cup \mathcal{R}^{-1}$ is symmetric. (I can give a proof of this, if you would like.) I wanted to ...
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31 views

Can someone verify my work for finding the following closures?

This is the problem I am currently working on Let R be the relation on the set {0, 1, 2, 3} containing the ordered pair (0,1), (1,1), (1,2), (2,0), (2,2), and (3,0). Find the a.reflexive closure of ...
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15 views

Can someone verify my work for finding the following relations?

I am working on this problem Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs {(1,1), (1, 2), (1,3), (2,3), (2,4), (3,1), (3,4), (3,5), (4,2), (4,5), (5,1), (5,2), ...
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38 views

Is my answer for the composite relation correct and not the textbook's?

Note: This example is from Discrete Mathematics and Its Applications [7th ed, example 5 pg 593] Here is how my textbook's way of representing a relation with a matrix And the definition of a ...
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50 views

Equivalence Relations with equivalence classes

Consider the following relation on $\mathbb{Z} \times \mathbb{Z}$: $(x, y)\sim (x', y')$ iff $xy' = x'y$. (a) Prove that it is an equivalence relation. (b) Consider the set of equivalence classes ...
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34 views

xor of three relations using Relation Algebra operations

Suppose I have three relations R1, R2, and R3. How can I specify xor of these three relations using relation algebra operations. How this scales up (for example, for four relations)? Thanks I add ...
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26 views

Partial and total orders

From Exercise 4.4.9 of How To Prove It: Suppose $R$ is a partial order on $A$ and $S$ is a partial order on $B$. Define a relation $L$ on $A \times B$ as follows: $L = \{((a, b), (a', b')) \in (A ...
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44 views

Reflexive, symmetric and transitive closure of the following graph

First of all, I have to understand which relation this graph represents: Since there are 4 different elements, suppose there's the set $A = \{ a, b, c, d\}$ The relation drawn above, I think, can ...
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57 views

Real life example of relations with various combination of properties

Attempted a set of questions as below: ...
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34 views

Does R(5,7) hold or not in this relation?

This is the question: Let $A=\{1,...,7\}$ and let $R\subseteq A\times A$ be a relation which is symmetric and transitive. You have been given some partial information about the relation which is ...
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44 views

Inverse Relation of Irreflexive Property.

We are taking the inverse of relation to check that inverse of R is transitive, reflexive , symmetric and anti-symmetric to as it is on R (not inverse).. My question is that why we are not taking the ...
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54 views

Finding Equivalence Classes for Infinite Sets

Let R be the relation on the set of rational numbers Q defined as follows: for all q, r ∈ Q, qRr iff q − r ∈ Z, where Z is the set of integers. R is an equivalence relation on Q. What is the ...
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Finding Relations algebraically

I have selfstudy on this subject and want to know if I am grasping the concept well. Here is the question: Let $A=Z^+$, all integers that are positive; Let $R$= relation defined by $aRb$ iff there ...
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35 views

Suppose A and B are sets. If $|A| = m$ and $|B| = n$, then how many relations are there from $A$ to $B$?

Coudn't I say that it is $m\times n$ relations because the collection of all ordered pairs is the Cartesian Product $A\times B$?
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Uniqueness of smallest element in poset

Prove that a smallest element, if it exists, is determined uniquely. This is follows directly for definition. An element $a \in X$ is called the smallest element of $(X, \preceq)$ if for every $x \in ...
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38 views

Properties of relations on Z

$$R_3 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|\frac{x-y}{5} \in \mathbb{Z}\} $$ $$R_4 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|x > y \} $$ I need to know which one is reflexive, symmetric, ...
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35 views

A partial order with more properties than would be expected

Consider the relation: $$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$ This is usually used for defining the (positive) fractions $\Bbb ...
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94 views

Have you seen this property of tolerance relations before?

Let $A$ be a set equipped with a binary reflexive and symmetric relation $\uparrow$ (such relations are often called tolerances, see also "Are there real-life relations which are symmetric and ...
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68 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
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52 views

Hasse Diagrams - Relations

I have the following question: Draw the Hasse diagram for the following partially-ordered set: The relation $X$ is a subset of $Y$, on the set $\{ \{0\}, \{2\}, \{0,1\}, \{0,2\}, \{2,4\}, ...
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33 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 ...
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177 views

General (Set Builder) definition for a relation composed with itself n times

Questions What does the set builder notation for $S\circ R$ look like? I'm having the most trouble knowing when there is too much information or not enough information on either side of the 'such ...
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36 views

Prove the following $(R\cap S)^n=R^n \cap S^n$

I would like to prove the following without induction. $$(R\cap S)^n=R^n \cap S^n$$ We can start by take $(a,b)\in (R\cap S)^n$ its represent a path from $a$ to $b$ right? Any hints? Thanks.
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70 views

Not closed under equality?

One of the Peano axioms state that "For any $a \in \Bbb N : a = b, b \in \Bbb N$." An example where transition is not closed under equality is the relation to friends. C may not be B's friend, but A ...
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31 views

Relation which is only locally a function

Is there a term for a relation which is not a function (because it maps multiple inputs to the same output), but which looks like one locally? That is, for any $\langle x,y\rangle\in R$, there's some ...
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46 views

Preordering on a set

I am given a definition which states that a 'preodering on a set is a relation that is reflexive and transitive.' Show that a relation $\leq$ defined on $\mathbb{C}$ by $z_1 \leq z_2$ iff $|z_1| = ...
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108 views

Prove that for any $W\in Q$, there exist a bijection $f:[0]\to W$

Suppose $R$ be the relation on $[0,1)$ s.t. $aRb\iff a-b$ is rational. Let $[0]$ be the equivalence class with respect to relation $R$ on $[0,1)$. Let $Q$ be the set of all equivalence class on [0,1). ...
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90 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 ...
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152 views

Example relations: pairwise versus mutual

There are by now several questions on math.se asking about pairwise versus mutual relations, eg: • When does “pairwise” strengthen and when does it weaken? • Relation: pairwise and mutually • ...
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109 views

Total order and $\inf,\min,\max,\sup$ of set?

I have a relation $\mathbb{R\subseteq M\times M}$, which is not even a partially order : $$ M=\{x\in\mathbb{R}:-3\leq x\leq3\}, R=\{(x,y):x>y\}. $$ Well what I'm trying to do is to make this ...
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220 views

Partially ordered set proof

I'm trying to proof if the following Relations R ⊆ M×M total order or partially order are. $M = \{1,2,3\} , R = \{(x,y) : x|y\}$ $M = {\bf Z} , R = \{(x,y) : x\vert y\}$ $M = {\bf N}, R = \{(x,y): y ...