Determine whether the set is closed for the given operations in exercises 25 and 26. Explain your answer. If not closed give an example to show why not.
a. The set of negative integers for multiplication
b. The set of negative integers for addition
Exercise 25:
a. The set of negative integers for multiplication is not closed. This is because when we multiply two negative integers, we get a positive integer. For example, -2 * -3 = 6, which is a positive integer. Thus, the set of negative integers for multiplication does not include all possible results of the operation and is therefore not closed.
Exercise 26:
b. The set of negative integers for addition is closed. This is because when we add two negative integers, we always get another negative integer. For example, -2 + (-3) = -5, which is also a negative integer. Thus, the set of negative integers for addition includes all possible results of the operation and is therefore closed.