Question

From the infinite set {1, 2, 3, ...}, remove the infinite set {2, 4, 6, ...}. What is the remaining set, and how large is it?

a) {1, 3, 5, 7, ...}, countably infinite
b) {1, 2, 3, 4, 5, 6, ...}, countably infinite
c) {2, 4, 6, 8, ...}, countably infinite
d) {1}, finite

Answer 1

After removing all even numbers from the set of all positive integers, the remaining set is the odd numbers {1, 3, 5, 7, ...}, which is countably infinite.

Step-by-step explanation:

When you remove the set of all even numbers {2, 4, 6, ...} from the set of all positive integers {1, 2, 3, ...}, you are left with the set of all odd numbers. The odd numbers can be represented as {1, 3, 5, 7, ...}, and this is a pattern that continues indefinitely. Thus, the remaining set is countably infinite, meaning that even though it does not end, you can count the elements in the sequence as they have a specific order and can be matched one-to-one with the set of natural numbers.