Final answer:
The identity elements for symmetric difference, intersection, and union are correctly identified, but the relative complement does not have the universe as its identity; the empty set is the identity for the relative complement operation.
Step-by-step explanation:
The student is asking about the identity elements for different set operations on the power set of a universe. An identity element is an element in a set that, when used in an operation with any other element of the set, leaves the other element unchanged. For the four set operations mentioned:
- The symmetric difference has the empty set as its identity because the symmetric difference of any set with the empty set results in the original set.
- The intersection has the universe as its identity because any set intersected with the universe itself will result in the original set.
- Union has the empty set as its identity because any set united with the empty set remains unchanged.
- Relative complement doesn't have the universe as its identity. Instead, the empty set is the identity for the relative complement operation because the relative complement of a set with the universe is the empty set.
Therefore, option d. Relative complement has the universe as its identity, is NOT true when considering binary operations on the power set.