Final answer:
To determine if a set is closed under an operation, we need to see if the operation produces elements that are also in the set. Set (a) is closed under addition mod 16, set (b) is not closed under addition mod 15, set (c) is closed under multiplication mod 15, and set (d) is not closed under multiplication mod 9.
Step-by-step explanation:
To determine if a set is closed under an operation, we need to see if the operation produces elements that are also in the set.
(a) {0, 4, 8, 12} addition mod 16:
The set is closed under addition mod 16 because if we add any two elements in the set and take the result mod 16, we will still get an element in the set.
(b) {0, 4, 8, 12} addition mod 15:
The set is not closed under addition mod 15 because if we add 8 and 12, the result is 20, which is not in the set.
(c) {1, 4, 7, 13} multiplication mod 15:
The set is closed under multiplication mod 15 because if we multiply any two elements in the set and take the result mod 15, we will still get an element in the set.
(d) {1, 4, 5, 7} multiplication mod 9:
The set is not closed under multiplication mod 9 because if we multiply 4 and 5, the result is 20, which is not in the set.