1. (25 points) For each of the ﬁve following binary relations R on the set A, determine whether

it has each of the four following properties, and brieﬂy justify your answer. Properties: i. R is reﬂexive. ii. R is symmetric.

iii. R is antisymmetric.

iv. R is transitive. Relations: (a) A is the positive integers and R is the set of pairs (m, n) such that every prime number

that divides am. also divides n and every prime number that divides 'n. also divides m. (b) A is all subsets of the natural numbers, that is, A = ‘P(N), and (S, T) E R if and only if

S' Q T. (c) A is the set of all ﬁnite sequences of lower case letters of the English alphabet (which

are not necessarily words, for example: app, apple, zzyzz) and R is the set of all pairs

(at) such that s aé t and a would appear before If in a dictionary. So (app, apple) 6 R

but (zzyzz, zyz) 9! R. (d) A is all subsets of the nonnegative integers, that is, A = 39(N), and R is the set of pairs

(3,5?) such that there exists a bijection f : S —> T. (Hint: you may assume that if

f : S —> T is a bijection, then its inverse function f ‘1 : T —> S is also a bijection, and

also that the composition of two bijections is a bijection.) (e) A is the set N of all nonnegative integers and R is the set of pairs (m,n) such that

m2 > n.