Problem 1: A bijection is an injective (one to one), surjective (onto) map between sets. If S = (0,1) and T = R, find a map from S to T which is (a) An injective but not surjective map, (b) a surjective but not injective map, (c) a bijection.
Problem 2: Which of the following is an equivalence relation on the set S?
a. a ~ b in S = Z if ¦a¦ = ¦b¦.
b. a ~ b in S = C if ¦a¦ = ¦b¦.
c. a ~ b in S = if arga - argb where if a Î C, a = reiq then ¦a¦= r and arga = q.
d. a ~ b in S = N if a - b divides n.
For the equivalence relations, what is the corresponding partition of S into equivalence classes? For the relations which are not equivalence relations, which of the three properties of equivalence relations fails?
Problem 3: If n is a positive integer, let Z*n be the subset of integers in (1,...., n) which are relatively prime to n. show that is a, b Î Z*n.