explain whether each set is closed under the given operation {-1,0,1}multiplication?

To determine whether a set is closed under a given operation, we need to check if applying the operation to any two elements in the set always results in another element in the set.

In this case, the set is {-1, 0, 1} and the operation is multiplication.

Let's analyze the set under multiplication:

1. Multiply -1 and -1: (-1) * (-1) = 1. 1 is not in the set {-1, 0, 1}, so the set is not closed under multiplication because (-1) * (-1) is not an element of the set.

2. Multiply -1 and 0: (-1) * 0 = 0. 0 is in the set {-1, 0, 1}, so the set is closed under multiplication for this operation.

3. Multiply -1 and 1: (-1) * 1 = -1. -1 is in the set {-1, 0, 1}, so the set is closed under multiplication for this operation.

4. Multiply 0 and 0: 0 * 0 = 0. 0 is in the set {-1, 0, 1}, so the set is closed under multiplication for this operation.

5. Multiply 0 and 1: 0 * 1 = 0. 0 is in the set {-1, 0, 1}, so the set is closed under multiplication for this operation.

6. Multiply 1 and 1: 1 * 1 = 1. 1 is in the set {-1, 0, 1}, so the set is closed under multiplication for this operation.

Based on our analysis, we can conclude that the set {-1, 0, 1} is closed under multiplication for some elements, but not all.