= Solution
The <relation powers> are $R^2=\{(a,c),(b,d)\}$, $R^3=\{(a,d)\}$, and $R^i=\varnothing$ for $i\geq4$. For $k=1$, $(a,b),(b,c)\in T$ but $(a,c)\notin T$, so <transitivity> fails. For $k=2$, $(a,c),(c,d)\in T$ but $(a,d)\notin T$, so it still fails.
For $k=3$, $T$ consists of every pair in which the first endpoint precedes the second in $a,b,c,d$. Composing any two such pairs again gives such a pair, so $T$ is <transitive>. Thus \b[the smallest $k$ is $3$]; this $T$ is the <transitive closure of a relation> $R$.
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