Question

Which of the given sets of whole numbers is closed under addition? If the set is not closed, give an example of two elements from the set whose sum is not in the set.

(a) {10, 15, 20, 25, 30, 35, 40, ...}
(b) {1, 2, 3, . . . , 1000}
(c) {0}
(d) {1, 5, 6, 11, 17, 28, ...}
(e) n ≥ 19
(f) {0, 3, 6, 9, 12, 15, 18, ...}

Answer 1

Sets (a), (c), (e), and (f) of whole numbers are closed under addition because the sum of any two elements in each set remains in the set. Sets (b) and (d) are not closed under addition as their elements can sum to numbers outside of the sets.

Step-by-step explanation:

The question is asking which sets of whole numbers are closed under addition. A set is closed under addition if the sum of any two elements in the set is also an element of the set.

1. Set (a) consists of multiples of 5 starting with 10. It is closed under addition because the sum of any two multiples of 5 is also a multiple of 5, and thus within the set.
2. Set (b) {1, 2, 3, ... , 1000} is closed under addition for numbers that sum to 1000 or less. For example, 500 + 500 = 1000 is in the set, but 1000 + 1 = 1001 is not, so it is not closed under addition.
3. Set (c) contains only the number 0, which is closed under addition since 0 + 0 = 0.
4. Set (d) is not necessarily closed under addition as there is no clear pattern that guarantees closure.
5. Set (e) consists of all natural numbers 19 and greater. This set is closed under addition since the sum of any two such numbers is at least 19.
6. Set (f) consists of multiples of 3. It is closed under addition because the sum of any two multiples of 3 is itself a multiple of 3, and thus the sum is within the set.