for example: computer science, economics, psychology, ematica will do computations in the symmetric group. So will SAGE. See Beezer's SAGE exercises in Judson [50]. Disjoint Cycle Notation for Permutations. Let us define the cycle notation by example. Consider the permutation = 2 3 45 5) 5 3 4 2 1 This permutation sends 1 to 5 and 5 to 1. In addition, it sends 2 to 3, 3 to 4, and 4 to 2. So we write =(15)(234)=(234)(15). The cycles (15) and (234) are said to be disjoint since each acts on disjoint sets of num- bers, namely {1,5} and {2, 3, 4}. Since the cycles (15) and (234) act on disjoint sets of numbers, the cycles commute: that is, (15)(234) = (234)(15). Exercise 3.1.1 Find the disjoint cycle decomposition for every permutation in S3 and S4. 2. Let S = {0, 2, 4, 6}, and T = {1, 3, 5, 7}. Determine whether each of the following sets of ordered pairs is a function from S to T. If so, is it injective, surjective, and bijective? a. {(0, 2), (2, 4), (4, 6), (6, 0)} b. {(6, 3), (2, 1), (0, 3), (4, 5)} c. {(2, 3), (4, 7), (0, 1), (6, 5)}