1) Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) $\displaystyle E$ R if and only if

a) x + y = 0.

b) x = ±y.

c) x - y is a rational number.

d) x = 2y.

e) xy >= 0.

f) xy = 0.

g) x = 1.

h) x = 1 or y = 1.

2) Let R1 = {(1, 2), (2, 3), (3, 4)} and R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find R1 - R2.

Would the answer of # 2 be Ø or {Ø} ?

3) Show that the closure of a relation R with respect to a property P, if it exists, is the intersection of all the relations with property P that contain R.

4) Use Algorithm 1 to find the transitive closures of these relations on {a, b, c, d, e}.

a) {(a, b), (a, c), (a, e), (b, a), (b, c), (c, a), (c, b), (d, a), (e, d)}

Thank You very much for your help!

Anu