Questions tagged [relations]

Question 1

First, some preliminary definitions. A serial relation is a binary relation $R$ on a set $S$ where this property holds: $(\forall x)(\exists y)xRy$, where the quantifiers range over $S$. Now, let $*$ ...

Question 2

A binary relation $R$ is said to be cycle-free if there are no cycles in the relation, meaning, for every positive integer $n$, there are no $x_1,...,x_n$ such that $x_1Rx_2...Rx_1$. Also, a strict ...

Question 3

I'm reading Invitation to Discrete Mathematics (2nd edition) by Matousek and Nesetril. Page 41, problem #2 asks: Prove that a relation $R$ on a set $X$ satisﬁes $R ◦ R^{-1} = ∆X$ if and only if $R$ ...

Question 4

This is a segue from this question I posted the other day. For $a, b:\mathbb N\to\mathbb R$, we write $a\prec b$ iff $\{n\in\mathbb N:a_n<b_n\}$ is cofinite. It is easy to see $\prec$ is a strict ...

Question 5

For $a, b:\mathbb N\to\mathbb R$, we write $a\prec b$ iff $\{n\in\mathbb N:a_n<b_n\}$ is cofinite. It is easy to see $\prec$ is a pre-order relation over $\mathbb R^\mathbb N$. Let $$\begin{aligned}...

Question 6

Given is a quadrilateral ABCD. Its diagonals intersect at point O. It is given that: AO = CO and: angle(ADC) > angle(ABC) From the given data, what can we deduce about the relationship between DO ...

Question 7

There are a couple questions on this site that seem to be asking "the same" question, but I'm asking about the category theory version. Let $\mathcal{C}$ be a lextensive category. A ...

Question 8

A few days ago my professor told us that $ \tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi $ then xyz = x+y+z (we proved it). He asked us to find all the possible natural numbers that satisfy the relation $...

Question 9

A binary relation $R$ over a set $S$ is said to be rigid (respectively, strongly rigid) iff the structure $(S;R)$ has no nontrivial automorphisms (respectively, no nontrivial endomorphisms). The rigid ...

Question 10

Fix two partial orders, $R$ and $R'$, on the same set $X$. Denote by $F(R)$ and $F(R')$ the set of all filters of $R$ and $R'$ respectively. Specifically, for $f \subseteq X$ we have $f \in F(R)$ if $...

Question 11

(I do not know whether the definitions below already have names in the literature; apologies in advance.) Let $R \subseteq \mathcal{X}\times\mathbb{N}$ be a binary relation. We declare that $A \...

Question 12

Let $1 \leq i \leq N$ and $1 \leq j \leq M$ denote two indices. Let $R$ denote a relation between $i$ and $j$, such that $R(i,j) = 1$ if $i$ and $j$ are related and $R(i,j) = 0$ otherwise. I have a ...

Question 13

Let suppose that we have test with $2$ sections: $X$ and $Y$. Each section can be evaluated as some element from $\{A, B, C\}$, where $A$ is better than $B$ and $B$ is better than $C$. Final mark is ...

Question 14

Let $(X,\mathcal{A})$ be a measurable space, and let $\{\le\}$ be a partial order relation on $X$ such that, seen as a subset of $X\times X$, is measurable for the product $\sigma$-algebra. (This has ...

Question 15

I am trying to understand the proof of the following equivalence: The proof is this: My doubt is: Why antisymmetry implies $a\neq c$? The version of antisymmetry I have is $(x\leq y) \wedge (x\geq y) \...

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Questions tagged [relations]

Is every serial relation a generalized divisor relation?

Is the class of cycle-free binary relations the same as the class of binary relations whose transitive closure is a strict partial order?

Reflexive and Antisymmetric relation question

Are $(c_0^+, \prec)$ and $(c_0^+, \succ)$ isomorphic?

Are $\ell^1_+$ and $\ell^\infty_+$ order-isomorphic?

Geometry - Quadrilateral diagonals such that one divides the other into equal parts

If a complemented relation is symmetric, is its complement symmetric?

Three natural numbers whose sum is equal to their product. [duplicate]

Does the rigid relation principle imply the strongly rigid relation principle over ZF?

Ordering the set of all filters on a partial order.

2 disjoint subsets of $\mathbb{N}$ which cover two binary relations from $\mathbb{R}$ to $\mathbb{N}$

Inverting the order of a constrained double-summation

Function that describes ordered relation

When do down-sets in a measurable order generate the $\sigma$-algebra?

Why antisymmetry implies $a\neq c$?

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