0

I'm trying to remember a shorthand for a binary relation on relations: Suppose relation $R_2$ contains every tuple that is in $R_1$, and at least one additional tuple. Do we say "$R_2$ is stronger than $R_1$," or is there some other term that should be used?

Norman Ramsey
  • 143
  • see here. This book does use "stronger" / "weaker" to compare relations while it uses "finer" / "coarser" to compare partitions. – JMoravitz Feb 01 '22 at 17:38
  • 1
    Interestingly though, that cited passage indicates that it $S$ is stronger than $R$ iff $S\subset R$, so the meaning you wish to apply to "stronger" vs "weaker" is flipped. In terms of analogy, "Two people live in the same city" is a stronger piece of information to know than "Two people live in the same country." As such, in your example we should have said $R_2$ is weaker than $R_1$. – JMoravitz Feb 01 '22 at 17:43
  • And that passage provides a definition only for equivalence relations. – Norman Ramsey Feb 01 '22 at 18:14
  • Read further. "2.10 Definition: Let $R$ and $S$ be two binary relations on a set $X$. Then $S$ is said to be stronger than $R$ (or $R$ is said to be weaker than $S$) if for all $x,y\in X,~xSy$ implies $xRy$." It stated binary relations here, it did not specify equivalence relations. – JMoravitz Feb 01 '22 at 18:18
  • @JMoravitz ah perfect, that's exactly what I was hoping for. Care to make an answer so I can upvote and accept? – Norman Ramsey Feb 01 '22 at 23:26
  • @JMoravitz BTW the reason I didn't spot it is that Google's preview doesn't show me those pages. I get page 164 and then the next page I am permitted to see is page 168, and they are up to Proposition 2.14. I wonder why you get to see those pages and I don't? – Norman Ramsey Feb 01 '22 at 23:28

1 Answers1

2

As noted in the comments, $R_1$ is stronger than $R_2$ if $x\,R_1\,y$ implies $x\,R_2\,y$.

Why people so persistently put answers in comments is something I'll never understand.

Norman Ramsey
  • 143