So I'm wondering about some finer points of relation algebra, a field in which I'm not an expert:

Given relations $A$, $B$ and $C$ over the same set $S$, is it always true that $A^*\subseteq (B\cup C)^*$ iff $(A\setminus B)^*\subseteq C^*$ ?

Here, $A^*$ denotes the reflexive, transitive closure of $A$, $\cup$ is set union, and $\setminus$ is set difference.


Consider $S=\{1,2,3\}$ and relations $A=\{(1,3)\}, B=\{(1,2)\}, C=\{(2,3)\}$. Then

$A^*=(A\setminus B)^*= A\cup\{(i,i)\ |\ i\in S\}$,

$(B\cup C)^* = \{(i,j)\ |\ i,j\in S, i\le j\}$, and

$C^* = C\cup \{(i,i)|i\in S\}$.

In particular, $A^*\subseteq (B\cup C)^*$, but $(A\setminus B)^*\nsubseteq C^*$.


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