Talk:Minimal WRT Restriction

Prime.mover, since you are an expert at logic, do you think you could formulate a clean generalization of this? If $B \subseteq A$, $R$ is a relation on $A$, and $P(B, R)$ is any statement that in some sense only uses $R$ to compare elements of $B$, then $P(B,R)$ is equivalent to $P(B,R \restriction B)$. --Dfeuer (talk) 19:55, 19 April 2013 (UTC)


 * Does this help? : Properties of Restriction of Relation --prime mover (talk) 20:01, 19 April 2013 (UTC)


 * Not really, although all of those proofs could be written to use the general principle I'm so vaguely describing. The intuitively trivial principle proves that (for suitable _____):


 * $R$ is _____ on $B$ iff its restriction to $B$ is _____.


 * The essential limitation is that it must be possible to write the propositional formula so that $R$ appears only within something of the form $xRy$, where $x$ and $y$ are required or known to be in $B$. It cannot have, for example, anything about the transitive closure of $R$, or the unrestricted image of $R$, or any such funny business. --Dfeuer (talk) 20:19, 19 April 2013 (UTC)


 * This is beginning to sound like a conversation from http://clientsfromhell.net/ ... --prime mover (talk) 20:42, 19 April 2013 (UTC)


 * Haha. It's not that bad. Some examples:
 * Let $R$ be a relation on $A$ and let $B$ be a subclass of $A$. Let $R'$ be the restriction of $R$ to $B$.
 * $R$ is transitive on $B$: the property is $\forall a,b,c \in B: aRb \land bRc \implies aRc$.
 * $\forall S \subseteq B: |R^{-1}(S) \cap B| < 3$. While this doesn't have the required form, it can be put in such a form by rewriting the inner expression as $|\{x \in B: \exists s \in S: xRs \}|$.


 * Non-example: Just about any statement involving transitive closure. If $A = B \cup \{ \infty \}$ and $R$ extends $R'$ by letting $xR\infty$ and $\infty R x$ for all $x \in B$, there won't be any meaningful relationship between $R^+ \restriction B$ and $(R \restriction B)^+$. --Dfeuer (talk) 21:13, 19 April 2013 (UTC)

I'm going to have to reiterate the aims of this website: it's not intended as a place for doing research, it's primarily a reference resource.

I expect there are plenty of sites (mathhelpforum might be a good place to go, if it's still going) which can help you out with this. As for me I expect I could figger out what you're about but I don't have the headspace at the moment to do much more than the ongoing maintenance task that I'm involved in. --prime mover (talk) 21:19, 19 April 2013 (UTC)


 * Research, eh? This is just formalizing intuition that we already use here without bothering (in most cases) even to mention it. The theorem this talk page is attached to is one I doubt most texts would even bother to prove—it's just too trivial. This sort of thing is really all over the place. --Dfeuer (talk) 21:41, 19 April 2013 (UTC)


 * My rate is $\$200$ an hour (payable via PayPal). --prime mover (talk) 21:54, 19 April 2013 (UTC)