Which pair of transitive relations gives a union that is not transitive?

Correct answer: R={(1,2)}, S={(2,3)}

Direct answer: A. A relation is transitive when two linked pairs (a,b) and (b,c) force (a,c). Each singleton relation in option A is itself transitive: R has only (1,2), and S has only (2,3), so neither relation contains two consecutive pairs requiring a further pair. However, their union contains (1,2) and (2,3), while it does not contain (1,3). Therefore the union fails transitivity. Option B gives {(1,1),(2,2)}, an identity-type relation, which is transitive. Option C gives the empty relation; transitivity is vacuously true because there are no starting pairs. Option D gives A×A, since adding the empty relation changes nothing; the universal relation is transitive. Common warning: transitivity need not pass from two relations to their union, because the first link may come from one relation and the second from the other.