Tagged Questions

2
votes
2answers
62 views

Is there an infinite sequence AB, BC, CD, DX, …, YZ

Is it possible to construct an infinite set of ordered pairs of form S = {(A, B), (B, C), (C, D), (D, x), ..., (y, Z)}? Every element (B, C...) must appear only once as the first object in one of the ...
1
vote
1answer
54 views

Operations and relations

To what extent do operations and relations overlap? Is there some more general structure that encompasses both of these things? Thanks
4
votes
2answers
80 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
102 views

Proving a relation between 2 sets as antisymmetric

Let $U = \{1,...,n\}$ And let $A$ and $B$ be partitions of the set $U$ such that: $\bigcup A = \bigcup B = U$ and $|A|=s, |B|=t$ Let's define a relation between the sets $A$ and $B$ as follows: $B ...
2
votes
1answer
103 views

A characteristic of intersection with cartesian product

Fix some binary relation $f$. Does there necessarily exist a set $C$ such that $(x\times x)\cap f\ne \varnothing \Leftrightarrow x\cap C\ne \varnothing$ for all sets $x$?
0
votes
1answer
97 views

$<$ on a preorder is a strict partial order

Definition: Suppose $X$ is a preorder. Define $x < y$ as $x \le y$ and $y \not\le x$ for each $x, y \in X$. Question: Show that this gives a strict partial order on $X$.
2
votes
1answer
131 views

Preorders, chains, cartesian products, and lexicographical order

Definitions: A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive. A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot ...
4
votes
2answers
111 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 ...
-7
votes
1answer
146 views

Order of products and order of multipliers

I asked this question (and have received an answer) at MathOverflow. Now a little more difficult question: Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is ...
0
votes
0answers
107 views

Provide examples of non-equivalence of certain set-theoretic concepts

A free star $S$ on a set $A$ is a collection of subsets of $A$ such that: $\emptyset \notin S$; $\forall X, Y \in \mathscr{P} A : X \cup Y \in S \Leftrightarrow X \in S \vee Y \in S$. Let $I$ is ...
-2
votes
1answer
106 views

Find an elegant proof of a set-theoretic equiality about relations

I am now attempting to prove the following theorem. I am in half-underway of the proof and it seems I can do it by myself. But the proof I am now constructing is not elegant. Could anyone provide a ...
0
votes
1answer
125 views

Construct a new relation from several relations

Let $f_0,\dots,f_{n-1}$ are relations, where $f_i$ is a relation of arity $m_i$ for every $i=0,\dots,n-1$. How to construct an $(m_0+\dots+m_{n-1})$-ary relation from them? For my previous erroneous ...
1
vote
2answers
135 views

Unnecessary property in definition of equivalence relation [duplicate]

Possible Duplicates: Symmetric, Transitive and reflexive Why isn't reflexivity redundant in the definition of equivalence relation? Dependence of Axioms of Equivalence Relation? Let ...
0
votes
0answers
99 views

Simplify a set-theoretic formula

Let $a\in\mathscr{P}\prod A$ is an $n$-ary relation (where $A$ is an $n$-indexed family of sets) and $f$ is an $n$-indexed family of functions. Can the formula defining the following predicate (for ...
-4
votes
1answer
295 views

A counter-example for a set-theoretic problem?

I have proved the below conjecture for the special cases $n\in\{0,1,2\}$. The cases $n\ge 3$ (finite and infinite) are unknown. If the following conjecture is true, I don't expect that you will be ...
1
vote
1answer
234 views

Empty set as a relation

The empty set is an $n$-ary relation for every $n$, right? How should we call a pair $(n;r)$ consisting of some number $n$ and an $n$-ary relation $r$? To specify $n$ is necessary only when $r$ is ...
-2
votes
1answer
87 views

Simplify a formula about relations

Let $F$ is an $n$-ary relation (with $n$ being any index set). Can the following formula be simplified? $$(\lambda x\in n:s(x))\in F$$ ($s$ is some function). Here $\lambda$ is defined as: ...
4
votes
3answers
227 views

Do there exist interesting binary relations satisfying reflexivity and symmetry, but not transitivity?

Given the usual set-theoretic definition of a binary relation[1], along with the usual notions of reflexivity symmetry transitivity Do there exist any interesting (i.e. surprising, yielding novel ...
5
votes
0answers
182 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 ...
1
vote
1answer
294 views

Prove something is a partial order

A relation $\mathrm{R}$ is defined on the set of all positive integers by: $x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$. Prove that $\mathrm{R}$ is a partial ...