Natural numbers are closed under subtraction.
Idea of “closed under subtraction”:
If a set is closed under an operation, then doing that operation with any two numbers from the set gives an answer that is still in the set.
Here the set is the natural numbers: \(\mathbb{N} = \{1, 2, 3, \dots\}\).
What we need to check:
Take any two natural numbers \(a\) and \(b\). Is \(a - b\) always a natural number?
Counterexample (one example is enough to say “not closed”):
Choose \(a = 3\) and \(b = 5\).
Compute: \(3 - 5 = -2\).
But \(-2\) is not in \(\mathbb{N}\).
Therefore:
Since we found natural numbers \(a\) and \(b\) where \(a - b\) is not natural, the set of natural numbers is not closed under subtraction.
Note: Even if someone includes \(0\) in \(\mathbb{N}\), we can still pick \(a < b\) (like \(3\) and \(5\)) and get a negative result, which is not natural.