Properties of relations: given a relation r on a set a:
reflexive - r is reflexive iff xrx for every x in a r is reflexive iff (x, x)  r for every x in a
symmetric – r is symmetric iff xry → yrx r is symmetric iff (x, y) r → (y, x)  r
antisymmetric – r is antisymmetric iff xry → y x or x = y r is antisymmetric iff (x, y)  r → (y, x) r or x = y
transitive – r is transitive iff xry and yrz → xrz r is transitive iff (x, y) r and (y, z) r → (x, z) r
prove or disprove each of the properties for the given relation.
4) a = {integers} r = { (x, y) | y ≥ 2x }
reflexive?
symmetric?
antisymmetric?
transitive?