The set of the first five positive integers whose squares are odd numbers is {1,3,5,7,9}.
We can verify this as follows:
12 = 1, which is odd an odd number
32 = 9, which is odd an odd number
52 = 25, which is odd an odd number
72 = 49, which is odd an odd number
92 = 81, which is an odd number
Therefore, the set of the first five integers whose squares are odd numbers is {1,3,5,7,9}.