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, ...}

Answer 1

a) {10, 15, 20, 25, 30, 35, 40, ...} is not closed under addition. b) {1, 2, 3, ..., 1000} is closed under addition. c) {0} is closed under addition. d) {1, 5, 6, 11, 17, 28, ...} is not closed under addition.

Step-by-step explanation:

In order for a set of whole numbers to be closed under addition, the sum of any two numbers in the set must also be in the set. Let's analyze each given set:

a) {10, 15, 20, 25, 30, 35, 40, ...}: This set is not closed under addition because if we take 10 and 15, their sum is 25, which is not in the set.

b) {1, 2, 3, ..., 1000}: This set is closed under addition because if we take any two numbers from the set and add them, their sum will still be in the set.

c) {0}: This set is closed under addition because if we take 0 and add it to any other number in the set, the sum will still be 0.

d) {1, 5, 6, 11, 17, 28, ...}: This set is not closed under addition because if we take 5 and 6, their sum is 11, which is not in the set.