A relation R on a set A is called irreflexive if for all a ∈ A, (a, a) ∉ R.
(a) Give an example of a relation R on Z where R is irreflexive and transitive but not symmetric.
(b) Let R be a nonempty relation on a set A. Prove that if R satisfies any two of the following properties - irreflexive, symmetric, and transitive - then it cannot satisfy the third.
(c) If | A | = n ≥ 1, how many different relations on A are irreflexive? How many are neither reflexive nor irreflexive?