# Questions tagged [relations]

For questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric, etc), a composition of relations, or anything else concerning a relation on a set.

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### Making a reflexive and transitive relation into a partial order

$R$ is a reflexive and transitive binary relation with field $A$. Prove that equivalence relation $S$ in $A$ exists and partial ordering $T$ in $A/S$, such that for arbitrary $x$ and $y$ from $A$ the ...
96 views

### An equation with arbitrary binary relations

Let $f$, $g$, and $b$ are binary relations (on some set $\mho$). Let the predicate $F$ be defined by the formula $F(a)\Leftrightarrow (a\circ f^{-1})\cap (g^{-1}\circ b)\ne\emptyset$ for every binary ...
402 views

### Generating a minimal transitive relation containing a given collection of transitive relations

Suppose I have a collection $U$ of transitive binary relations on an arbitrary set $A$; elements of $U$ are subsets $S$ of $A \times A$ such that if $(a,b) \in S$ and $(b,c) \in S$ then $(a,c) \in S$ ...
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### Find the number of binary relations.

Let $X$ = {$a,b,c,d,e$}. Let us call a binary relations $R$ on $X$ special if it satisfies all of the following conditions: (i) $R$ is reflexive, (ii) $R$ is symmetric and (iii) $R$ contains the pair (...
131 views

### About function inj, surj and something else. Is this exercise resolved correctly?

This is my problem: For every couple of integers $(a,b)\in\mathbb{Z}\times(\mathbb{Z}\backslash\{0\})$ we denote with $r(a,b)$ the remainder of the division between $a$ and $b$. Consider the ...
179 views

### Hints needed on basic proof involving functions and relations.

Let $F = \{f\mid f\colon \mathbb R \to \mathbb R\}$, and define a relation $S$ on $F$ as follows: $S = \{(f,g) ā F \times F \mid \exists h \in F :f = h\circ g\}$. Let $f$, $g$ and $h$ be the functions ...
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### directed graph representing the inverse relation

Let $R$ be a relation on a set $A$. Explain how to use the directed graph representing $R$ to obtain the directed graph representing the inverse relation $R^{-1}$ ($R$ inverse).