Learn Prove which operations preserve relation axioms, repair failed closure claims, and derive short directed paths from an extremal argument.
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Closure proofs must check each axiom separately.
No. On three distinct elements, include both directions between a and b and between b and c, but omit the pair from a to c. This relation is symmetric but not transitive.
The tournament theorem bounds path length rather than requiring adjacency.
No. A walk from u through w to v uses the two edges u to w and w to v. A direct edge from u to v is a different, one-edge witness.
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Compose the rules and prove the reachability claim
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Use witnesses and complete finite counterexamples to test inverse, intersection, union and composition of relations.
6 practice tasks.
Use complete pairwise orientation to prove a two-step reachability theorem, then distinguish its sufficient condition from a characterization.
6 practice tasks.