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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39
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4answers
6k views

Why isn't reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ ...
6
votes
5answers
12k views

Number of relations that are both symmetric and reflexive

Consider a non-empty set A containing n objects. How many relations on A are both symmetric and reflexive? The answer to this is $2^p$ where $p=$ $n \choose 2$. However, I dont understand why this is ...
11
votes
4answers
5k views

Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
8
votes
2answers
12k views

If a relation is symmetric and transitive, will it be reflexive? [duplicate]

Possible Duplicate: Why isn't reflexivity redundant in the definition of equivalence relation? We had a heated discussion in class today and i still cant be sure if the professor was ...
97
votes
15answers
21k views

Are there real-life relations which are symmetric and reflexive but not transitive?

Inspired by Halmos (Naive Set Theory) . . . For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have ...
2
votes
1answer
4k views

Number of reflexive relations defined on a set A with n elements

Problem: If a set $A$ has $n$ elements in it, how many reflexive relations can be defined on it? My solution Is the answer ...
0
votes
3answers
226 views

Two questions about equivalence relations

Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6 $. Show that $\sim$ is an equivalence relation. This is what I tried: Reflexive: $x^2\pmod6 = x^2$ implying $x\sim x$ ...
2
votes
3answers
489 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 ...
1
vote
1answer
322 views

Direct products in the category Rel

Please describe direct products in the category Rel.
3
votes
2answers
5k views

How to find the number of anti-symmetric relations?

I know that given a set $A = \{1, 2, 3, ... , n\}$, the total number of relations on $A$ is $$2^{n^2}$$ The number of reflexive relations is $$2^{n^2 - n}$$ The number of symmetric relations is ...
2
votes
2answers
3k views

Proving this relation is transitive

Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$? I tried to ...
1
vote
1answer
67 views

Define a relation and find its equivalence classes.

Define a relation $\sim$ on $\Bbb{N}$ as follows. For any $a,b∈\Bbb N$, $a\sim b$ if and only if $ab$ is a perfect square. Show that $\sim$ is an equivalence relation. What are the equivalence ...
2
votes
1answer
209 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 ...
2
votes
2answers
984 views

Is statistical dependence transitive?

Take any three random variables $X_1$, $X_2$, and $X_3$. Is it possible for $X_1$ and $X_2$ to be dependent, $X_2$ and $X_3$ to be dependent, but $X_1$ and $X_3$ to be independent? Is it possible ...
1
vote
0answers
167 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 • ...
14
votes
3answers
1k views

Sole minimal element: Why not also the minimum?

A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$. The minimum (at most one) of a partially ordered set $S$ is an ...
3
votes
2answers
784 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...
3
votes
2answers
13k views

How to check whether a relation is transitive from the matrix representation?

$$\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}$$ This is a matrix representation of a relation on the set $\{1, 2, 3\}$. I have to determine if this relation matrix is ...
4
votes
3answers
2k views

Why is $y = \sqrt{x-4}$ a function? and $y = \sqrt{4 - x^2}$ should be a circle

So why is it a function, even though for example $x = 8$; you'll have $y = +2$ and $y = -2$. It'll fail the vertical line test. But every textbook considers it as a function. Did I misunderstand ...
9
votes
2answers
510 views

Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section "Questions." INTRODUCTION After receiving an answer to this question about extending the definition of ...
12
votes
2answers
359 views

When does “pairwise” strengthen and when does it weaken?

"Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap ...
12
votes
2answers
475 views

Can we extend the definition of a continuous function to binary relations?

Let $X,Y$ be topological spaces. A function $\phi:X\to Y$ is continuous iff for any open subset $A\subseteq Y,$ the preimage $\phi^{-1}(A)$ is open in $X.$ We could similarly define a relation ...
6
votes
5answers
23k views

Antisymmetric Relations

Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric? $$R = \{(1, 2), (2, 3), (3, 4)\}$$ Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then ...
3
votes
1answer
2k views

Prove that the empty relation is Transitive, Symmetric but not Reflexive

Question: Let $R$ be a relation on a set $A$. Prove that if $A$ is non-empty, the empty relation is not reflexive on $A$. the empty relation is symmetric and transitive for every set $A$. My ...
1
vote
1answer
67 views

Does $\neg(x > y)$ imply that $y \geq x$?

Given any arbitrary binary relation $\geq$ defined on some set $S$, we define a new binary relation $>$ on $S$ by: $$ x > y \quad\text{iff}\quad (x \geq y) \wedge \neg(y \geq x) $$ In accordance ...
6
votes
1answer
133 views

Let R be a relation on set A. Prove that $R^2 \subseteq R <=>$ R is transitive $<=> R^i \subseteq R ,\forall i \geq 1$

this is my first question here. I'm still relatively new to more advanced mathematics and don't have much experience with proofs yet. I'm self-studying at the moment and therefore have no one to check ...
5
votes
2answers
216 views

Counting non-isomorphic relations

On a set $X$ of $n$ elements, how many non-isomorphic relations are there? The number of relations on a set of $n$ elements is $|\mathcal{P}(X \times X)|=2^{n^2}$, but is there any way to give a ...
4
votes
2answers
132 views

Equivalence relations on classes instead of sets

Can someone please explain to me how to deal with equivalence relations on classes instead of sets? Is there some sort of generalisation of relations? Thank you
1
vote
2answers
4k views

Equivalence Relations and Inverse Relations

Prove that if $R$ is an equivalence relation on a set $A$, then $R^{-1}$ is also an equivalence relation on $A$. Solution: We know that $\forall R\in A$, $$ R = \{(a, b) | (b, a)\in A\} $$ ...
5
votes
1answer
297 views

Is this alternative definition of 'equivalence relation' well-known? useful? used?

I discovered that $$R \textrm{ is an equivalence relation on } A \;\equiv\; \langle \forall a,b \in A :: aRb \:\equiv\: \langle \forall x \in A :: aRx \equiv bRx\rangle \rangle$$ The nice thing about ...
5
votes
3answers
7k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
5
votes
2answers
886 views

Cardinality of relations set

I was thinking about cardinality of all symmetric relations, for example in $\mathbb{Z}$. I know, that if I have finite set (which contains $n$ elements), there are $2^{\frac{n(n+1)}{2}}$ symmetric ...
4
votes
1answer
188 views

The class of equivalence.

Let $Q$ be the following set of $\mathbb{Z}\times \mathbb{Z}$ \begin{align*} Q=\{(a,b)\in \mathbb{Z}\times \mathbb{Z} | b\neq 0\} \end{align*} Define the relation $\sim$ on $Q$ as ...
4
votes
2answers
154 views

Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$.

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$ I don't understand how I'm suppose to prove this: Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$ Also can you ...
4
votes
2answers
165 views

$<$ in a preorder

The author of the book I am studying defines $<$ for a poset as If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$. From this, I can ...
4
votes
2answers
499 views

The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
4
votes
3answers
188 views

What's the name for the equivalence induced by a function on its domain?

Any function $f$ with domain $X$ induces an equivalence relation on $X$, with classes $$\{f^{-1}(\{y\})\,:\, y \in \operatorname{im}f\;\} .$$ Is there a name for this equivalence? Thanks!
4
votes
5answers
4k views

Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$

Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$. (a) Prove that $R$ is an ...
3
votes
1answer
83 views

If $X,Y$ are equivalence relations, so is $X \times Y$

If $X,Y$ are reflexive, symmetric, and transitive, then $X \times Y$ is an equivalence relation where ${(a,b):a\in X, b\in Y}$. I am trying to self learn these topics. I do know what an ...
3
votes
0answers
45 views

A relation on 2 countable sets [duplicate]

Let $R$ be a relation on two countable sets $A$ and $B$, where $R\subset A\times B$, with the following properties: $\forall a\in A$ the set $\{b\in B: (a,b)\in R\}$ is finite. For any finite set ...
3
votes
4answers
366 views

Showing that $R$ is an equivalence relation on $X \times X$

Let $X = \{1,2,3,..,10\}$ define a relation $R$ on $X \times X$ by $(a,b)R(c,d)$ if $ad=bc$. Show that R is an equivalence relation on $X \times X$. I know that the $R$ have to be reflexive (because ...
3
votes
1answer
191 views

Relation: pairwise and mutually

Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions ...
2
votes
3answers
233 views

Is an Anti-Symmetric Relation also Reflexive?

According to the definition of an Anti-Symmetric Relation if xRy and yRx then x = y Which means, effectively, x is in relation with itself. Does this mean that anti-symmetry implies reflexive ...
2
votes
1answer
125 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
2
votes
3answers
1k views

How many reflexive binary relations there are on a finite countable set?

We know that binary relation is subset of Cartesian product made by set on to itself. let's say we have a set with two elements $A=\{0,1\}$ So Cartesian product is $C=A\times A = ...
2
votes
2answers
450 views

Proof of a Proposition on Partitions and Equivalence Classes

I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like, Proposition: If $\{A_i|i\in I\}$ is a partition of ...
1
vote
1answer
139 views

Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
1
vote
1answer
74 views

Two “adjunct” (quasi-inverse) functions

Let $A$, $B$ be fixed sets. What "means" the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ for functions $\alpha:\mathscr{P}A\rightarrow\mathscr{P}B$ and ...
1
vote
1answer
50 views

Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?

This is an exercise of an assignment I have: Suppose $A$ and $B$ are finite sets and $f\colon A\to B$ is surjective. Is it true that the relation “$|A| < |B|$” is a sufficient condition for ...
1
vote
1answer
96 views

Consider the relation R given by divisibility on positive integers that is xRy <-> x|y

Consider the relation R given by divisibility on positive integers that is xRy <-> x|y Is this relation reflexive? symmetric? anti-symmetric? transitive?? I understand it is reflexive and ...