If 10 is removed from the set of whole numbers, is the set closed with respect to addition? Explain why or why not.
Is the set closed under this operation?
5+5 = 10
(note that 5 ∈ℤ' and 10 ∉ℤ', where ℤ' is ℤ\10)
To determine if the set of whole numbers is closed with respect to addition after removing 10, we need to check if the sum of any two whole numbers will still give us a whole number.
Whole numbers are those numbers that start from 0 and include all positive integers (1, 2, 3, ...) and their negatives (0, -1, -2, -3, ...).
Let's take any two whole numbers, for example, 5 and 8. Their sum would be 13. Since 13 is still a whole number, it is not affected by removing 10 from the set.
Now, let's consider the case of 10 and 3. The sum of 10 and 3 is 13, which is also a whole number. So, even though we removed 10 from the set, the sum of 10 and 3 remains a whole number.
Therefore, we can conclude that removing 10 from the set of whole numbers does not affect the closure property with respect to addition. The set of whole numbers remains closed under addition because the sum of any two whole numbers will still result in a whole number.