a) Consider the following relation R on the set {x, y, z, v}:

R = (x, x),(x, y),(x, z),(x, v),(y, y),(y, z),(y, v),(z, z),(z, v),(v, v),(v, z) .

Determine whether R is reflexive, transitive, a partial order, and/or an equivalence relation.

b) Determine which of the following relations on R is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) R if and only if:

1. x + y = 0

2. x y is a rational number

3. x 6= y

4. x =