Binary Relations (transitivity)
A binary relation B on S is transitive provided that whenever there is a two-element “chain” in B there is also the corresponding “shortcut” in B.
(?x ? S, ? y ? S, ? z ? S)
(x, y) ? B and (y, z) ? B ? (x, z) ? B)
R8 = { (a, b), (a, c), (b, c), (c, c) } is transitive.
R9 = { (a, b), (b, a)} is not transitive, because
(a, b) and (b, a) form a chain, but (a, a), the shortcut, is not present.