For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive.
(a) R ⊆ Z+ × Z+ where a R b if a|b (read "a divides b," as defined in Section 4.3).
(b) R is the relation on Z where a R b if a|b.
(c) For a given universe U and a fixed subset C of U, define R on P(U) as follows: For A, B ⊆ U we have A R B if A⋂C = B⋂C.
(d) On the set A of all lines in R2, define the relation R for two lines ℓ1, ℓ2 by ℓ1 R ℓ2 if ℓ1 is perpendicular to ℓ2.
(e) R is the relation on Z where x R y if x + y is odd.
(f) R is the relation on Z where x R y if x - y is even.
(g) Let T be the set of all triangles in R2. Define R on T by t1 R t2 if t1 and t2 have an angle of the same measure.
(h) R is the relation on Z × Z where (a, b)R(c, d) if a ≤ c.