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)