Sets and Relations

Question

Describe, with proof, all relations R on {4, 5, 6} that are symmetric and transitive, and for which 4R5, but that are not equivalence relations.

Answer 1

Define aRb iff a-b≠0.

For {4, 5, 6}, 4R5=5R4, 5R6=6R5 (symmetry); 4R5 and 5R6⇒4R6 (transitivity), and 4R6=6R4 (symmetry). But 4R4=0, 5R5=0, 6R6=0 (non-reflexivity).

Therefore, even though the relations are transitive and symmetrical, they are not reflexive so equivalence does not apply.