Question

Suppose that every student at a university is assigned a unique 8-digit ID number. For i ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, define the set Ai to be the set of currently enrolled students whose ID number begins with the digit i. For each digit, i, there is at least one student whose ID starts with i. Do the sets A0, …, A9 form a partition of the set of currently enrolled students?

Answer 1

Yes, they form a partition

Explanation:

For a group of sets to be considered a partition of set A (currently enrolled students), all of the elements in set A must be contained in the subsets that form the partition and no individual element can be present in more than one subset.

In this situation, take set A0 for instance, all students whose ID numbers begin with 0 will be in this set and in this set alone. The same can be said for Ai where i ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Since no other possible value can be assigned for the first digit of an student ID, it is correct to affirm that sets A0, …, A9 form a partition of the set of currently enrolled students.