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Questions tagged [relations]

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.

128
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15answers
38k 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 the ...
63
votes
6answers
13k 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 $a$ ...
39
votes
3answers
3k views

How should I be avoiding this mistake? (To avoid missing solutions)

First of all, I am sorry if this is a question too simple or stupid. Consider the equation: $$ \log((x+2)^2) = 2 \log(5) $$ If I apply the logarithm law $ \log_a(b^c) = c \log_a(b) $ $$ \begin{...
22
votes
7answers
2k views

Proof that the empty set is a relation

In the book Naive Set Theory, Halmos mentions that the "The least exciting relation is the empty one." and proves that the empty set is a set of ordered pairs because there is no element of the empty ...
19
votes
3answers
3k 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 ...
19
votes
5answers
85k 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 $...
17
votes
4answers
14k views

Understanding equivalence class, equivalence relation, partition

I'm 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 ...
16
votes
3answers
61k views

Is my understanding of antisymmetric and symmetric relations correct?

So I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric ...
14
votes
4answers
2k views

What does it mean when I say that addition/multiplication for an equivalence relation is well defined?

I have trouble understanding this concept. Why is it necessary to prove that addition or multiplication is well defined in equivalence classes? My understanding of equivalence classes is that it must ...
14
votes
3answers
4k 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 ...
13
votes
3answers
45k views

Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same?

I don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have ...
13
votes
6answers
25k 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 ...
13
votes
5answers
273 views

Does “=” have to be interpreted as equality?

To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality? Let me ...
13
votes
2answers
716 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 $\rho\...
12
votes
10answers
6k views

I need a relation which is not reflexive, not symmetric, and not transitive

I need an example of a relation which is simultaneously not reflexive, not symmetric, and not transitive. Any accessible examples? Thanks in advance.
12
votes
4answers
10k views

Is any relation which contains only one ordered pair transitive?

I need clarification. Let $A=\{1,2,3\}$ be a set and $R=\{(1,2)\}$ be a relation on $A$. Is it a Transitive relation? I am confused because some text books say $R$ is transitive if it contains only ...
12
votes
4answers
54k views

Transitive Relations

For example, $$R = \{ (1,1),(1,2),(2,1),(2,2) \} \quad\text{for}\quad A = \{1,2,3\}.$$ This relation is symmetric and transitive. I understand that the relation is symmetric, but my brain does not ...
12
votes
4answers
38k 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 ...
12
votes
2answers
375 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 A_j\ne\...
11
votes
5answers
2k views

Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ implies ...
11
votes
3answers
1k views

What is the difference between Categories and Relations?

For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer. Category: A ...
11
votes
7answers
5k views

How do the Properties of Relations work?

This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - $...
11
votes
2answers
129 views

How many injective functions from $\{5,6,7,8,9\}$ to itself map $5$ and $6$ to another number?

How many injective functions from $\{5,6,7,8,9\}$ to itself map $5$ to any number other than $5$ and $6$ to any other number than $6$? Let $B=\{5,6,7,8,9\}, f:B\to B$. I think we can solve this by ...
11
votes
4answers
417 views

A problem about symmetric relations on finite sets.

We have these assumptions: $X$ is a finite set. $\sim$ is an irreflexive symmetric relation on $X$. for any subset $Y\subseteq X$ we define $$\mathcal{Cl}(Y)=\{A\subseteq Y\mid(\forall a,b\in A:a\ne ...
10
votes
5answers
1k views

Can a relation be transitive when it is symmetric but not reflexive? [duplicate]

Pretty much what the title asks. But here's some context: Suppose $X$ is finite and $R$ is a relation on $X$ that is not reflexive but it is symmetric. Also, suppose we can rule out $xRy$ and $yRz$ ...
10
votes
2answers
1k views

How do I define exactly what a function is?

While it is easy to understand what a function is intuitively, I've been trying to wrap my head around how to precisely define what a function is using only mathematical notation. My attempt at this ...
10
votes
4answers
9k views

Can a relation with less than 3 elements be considered transitive?

The generalize rule for a transitive relation is a -> b b -> c therefor a -> c If an element has less than 3 elements, can it still be transitive? If ...
10
votes
2answers
677 views

How is the relation “the smallest element is the same” reflexive?

Let $\mathcal{X}$ be the set of all nonempty subsets of the set $\{1,2,3,...,10\}$. Define the relation $\mathcal{R}$ on $\mathcal{X}$ by: $\forall A, B \in \mathcal{X}, A \mathcal{R} B$ iff the ...
10
votes
2answers
3k views

Main Theorems/Techniques for proving Homeomorphism?

General Question: what are the most common Theorems/Methods used to prove Homeomorphism? I encountered: - find the map explicitly - use the Compact-to-Hausdorff Lemma - find cts maps $f$ and $g$ s....
10
votes
2answers
880 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 ...
9
votes
3answers
1k views

Is every relation which is transitive and symmetric also reflexive?

I have seen a proof that every relation which is symmetric and transitive is also reflexive. if $A=\{1,2,3\}$ Then if $R=\{(1,2)(2,1)(1,1)\color{blue}{(2,2)}\}$ here $R$ is symmetric and transitive ...
9
votes
3answers
34k 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: {$(a,a)(a,b)(a,c)(a,d)(b,a)(b,...
9
votes
3answers
35k views

How Many Symmetric Relations on a Finite Set?

How many symmetric relations are there for an $n$-element set? Thank you.
9
votes
5answers
10k views

'Does not necessarily equal' symbol

What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in P:...
9
votes
2answers
27k 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 any good ...
9
votes
1answer
167 views

Counting number of mathematical objects and structures

Regarding the numbers of certain mathematical objects and structures, especially sets, relations and functions, I've compiled a list of the counts from various sources: Partitions of a set with $k$ ...
9
votes
2answers
165 views

Question from 'How to Prove It'

Below is the question from the book mentioned above: Suppose $f : A \rightarrow B$ and $R$ is an equivalence relation on $A$. We will say that $f$ is compatible with $R$ if $∀x \in A\forall y ∈ A(...
8
votes
4answers
3k views

Definition of smallest equivalence relation

I came across the term 'smallest equivalence relation' in the course of a proof I was working on. I have never thought about ordering relations. I googled the term and checked stackexchange and couldn'...
8
votes
3answers
22k views

Is the relation $R = \emptyset$ is it reflexive, symmetric and transitive ? Why?

Can someone help me understand the properties of the relation $R = \emptyset$ ? It looks to me like it's not reflexive, since there is no element related to any element, so the elements are not ...
8
votes
3answers
451 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 ...
8
votes
3answers
3k 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 ...
8
votes
4answers
5k views

Why is the empty set finite?

On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question): Definition 2.4: For any positive integer $n$, let $...
8
votes
4answers
581 views

Understanding Equivalence Classes?

I am reading about equivalence classes and I would like to make sure I understood thins properly. My book says: The set of equivalence classes under this equivalence relation [$\pmod n$] will be ...
8
votes
3answers
16k views

Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)

Antisymmetric: $\forall x\forall y[ ((x,y)\in R\land (y, x) \in R) \to x= y]$ reflexive: $\forall x[x∈A\to (x, x)\in R]$ What really is the difference between the two? Wouldn't all antisymmetric ...
8
votes
2answers
508 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 ...
8
votes
5answers
132 views

Finding $f(x)$ given a function $f : \Bbb N \to \Bbb N$ which satisfies $f\circ f(x) + f(x) = 2x+15$

I'm stuck on rewriting the function $f\circ f(x) + f(x) = 2x+15$ (into $f(x) = \dots)$. The answer given is $f(x) = x + 5$, which I can easily verify, but I do not know how to go about forming this ...
8
votes
1answer
713 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
8
votes
6answers
1k views

Interesting properties of ternary relations?

Many people are familiar with some properties of binary relations, such as reflexivity, symmetry and transitivity. What are the commonly studied properties of ternary (3-ary) relations? If you ...
8
votes
3answers
11k views

Reflexive Transitive Closure

The problem I am working on is, "Show that a finite poset can be reconstructed from its covering relation. [Hint:Show that the poset is the reflexive transitive closure of its covering relation.]" I ...
8
votes
1answer
285 views

How to prove an extension of ZFC is conservative

Working in ZFC. I've defined a function-like binary predicate $R$ on a proper class. It has to be recursive; i.e. $R(a,b)$ must usually depend on one or more $R(c,d)$ for some $c$s and $d$s ...