Articles of relations

Why is this relation not transitive but R = {(3,4)} is ?

While studying relations, I came across a strange question. Set $A=\{1,2,3,4\}$ on which the relation $R=\{(2,4),(4,3),(2,3),(4,1)\}$ is defined. It is said in the answer that the relation is not transitive. I am not able to find out why is it so. Let $R=\{(3,4)\}$. This is said to be transitive in the answer. Can anyone give […]

If $R=\{(x,y): x\text{ is wife of } y\}$, then is $R$ transitive?

If $R=\{(x,y): x\text{ is wife of } y\}$, determine whether the relation $R$ is transitive or not. My Try: For Transitivity, If $(a,b) \in R$ and $(b,c)\in R\;,$ Then $(a,c)\in R.$. Here If $x$ is a wife of $y$, then $y$ is not a wife of $z$. Therefore $x$ is not a wife of $z$. […]

Anti symmetrical relation

Currently learning about symmetrical and anti symmetrical relations. Been working on assignments but I cant seem to bend my head around this one, especially because I can’t understand the solution. What I thought an anti symmetrical relation is: a pair like $(x,y)$ that also exists as $(y,x)$ with $x = y.$ The question is: describe […]

Value assignment for complete, transitive relation on countably infinite set

Consider a set $A$ and a relation $r$. The relation $r$ is complete, i.e., for any $a,b\in A$, we have $arb$ or $bra$ or both. The relation $r$ is transitive, i.e., for any $a,b,c\in A$, if $arb$ and $brc$, then $arc$. Must there exist a function $f:A\rightarrow\mathbb{R}$ such that for any $a,b\in A$, we have […]

Problem about Hasse diagrams

Can someone help me to solve this problem. Are these Hasse diagrams lattices?

Does this category have a name? (Relations as objects and relation between relations as morphisms)

Given two relations $R\subseteq A\times B$ and $R’\subseteq A’\times B’$. Is it known/used that every relation $r\subseteq R\times R’$ can be characterized by two relations $\alpha\subseteq A\times A’$ and $\beta\subseteq B\times B’$ so that $((a,b),(a’,b’))\in r \iff \Big((a,a’)\in\alpha\wedge (b,b’)\in\beta\wedge (a,b)\in R\implies (a’,b’)\in R’\Big)$ and if $R”\subseteq A”\times B”$, $\,r’\subseteq R’\times R”$, where $r’$ is characterized […]

Maximum number of relations?

The question is that we have to prove that if $A$ has $m$ elements and $B$ has $n$ elements, then there are $2^{mn}$ different relations from A to B. Now I know that a relation $R$ from $A$ to $B$ is a subset of $A \times B$. And as the maximum number of subsets (Elements […]

$(a,b)R(x,y) \iff ay=bx$ is an equivalence on $\Bbb Z \times (\Bbb Z\setminus\{0\})$

This question already has an answer here: Proving this relation is transitive 2 answers

Trouble understanding equivalence relations and equivalence classes…anyone care to explain?

What exactly are equivalence relations and equivalence classes? The latter is giving me the most trouble; I’ve tried to read multiple sources online but it just keeps going over my head. Example question: What are the equivalence classes of 0 and 1 for congruence modulo 4?

Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict Orders

As I understand it, partial orders are binary relations that are: Reflexive Anti-symmetric Transitive An example would be $\subseteq$ for sets And if we add totality to this, we get a total (or linear) order, so a total order is Reflexive (this one is implied by totality, so can be removed from definition) Anti-symmetric Transitive […]