1
vote
1answer
50 views

Naturally definable order relations

Each and every function $f:X\rightarrow Y$ between two totally ordered sets $(X,\leq_X), (Y,\leq_Y)$ induces a relation $\preceq$ on $X$ by $$x \preceq x' :\equiv \begin{cases} x &\leq_X x' ...
3
votes
1answer
108 views

Deriving the Axiom of Infinity

I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ...
1
vote
1answer
70 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
2
votes
1answer
134 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
135 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
205 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
130 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 ...
-6
votes
1answer
203 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 ...
-2
votes
1answer
128 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 ...
1
vote
2answers
258 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 ...
-4
votes
1answer
439 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 ...
-2
votes
1answer
93 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: ...
8
votes
1answer
228 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
539 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 ...