For each relation given below determine whether it is reflexive, irreflexive or neither; symmetric, asymmetric, antisymmetric, or none of the above; transitive or not. Rom this determine whether it is a partial order, strict partial order, equivalence relation or none of the above.

a. The relationship of the natural numbers (a, b) ∈ R if and only if a = b + 1

b. The divides relation on N (a, b) ∈ R if and only if a|b

c. The divides relation on Z.

d. Given a function f (x) from the real numbers to the real numbers, let (a, b) ∈ R if and only if f (a) = f (b).