Suppose $R$ is a relation on $A$. Let $B = P (A) $\ $\{\varnothing\}$, and define a relation $S$ on $B$

Question

Suppose $R$ is a relation on $A$. Let $B = P (A) $\ $\{\varnothing\}$, and define a relation $S$ on $B$ as follows:

$S =\{ (X,Y)\in B \times B |\forall x \in X, \forall y \in Y,(xRy)\}.$

Prove that if $R$ is transitive, then so is $S$.

I'm stuck figuring out why is S not vacuously transitive in case we define $R = \{(1,2),(3,4)\}$ in order to prove it to be transitive

Answer 1

Statements of the form $\forall x \in \varnothing \colon ...$ are vacuously true, so if the empty set would be in $B$, then $(X, \varnothing), (\varnothing, X) \in S$ for any $X \in B$. So then if there are some $X, Y \in B$ with $(X, Y) \not\in S$ (we can always find such $X$ and $Y$ unless $xRy$ for all $x, y \in A$), then $S$ is not transitive since $(X, \varnothing), (\varnothing, Y) \in S$.

So it definitely is essential that $\varnothing \not\in B$ for the statement to be true.