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For a set to be closed under an operation, doing the operations on any two members of the set must result in another member of the set.
The set of even natural numbers is {2, 4, 6, ...}.
When two even numbers are added, the result is also an even number.
Adding two positive numbers results in a larger positive number.
Thus, adding two even natural numbers together results in another even natural number and so is closed.
The set of even whole numbers is {..., -6, -4, -2, 0, 2, 4, 6, ...}.
When two even numbers are added, the result is also an even number.
It does not matter which combination of positive and negative whole numbers are added as all positive and negative whole numbers (including zero) are in the set. Thus it is also closed under addition.
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Why are natural numbers not under addition?
Natural numbers are actually closed under addition. If you add any two if them, the result will always be another natural number.
Is there a subset of the natural numbers that is closed for addition?
Yes. The entire set of natural numbers is closed under addition (but not subtraction). So are the even numbers (but not the odd numbers), the multiples of 3, of 4, etc.
Which set of numbers is closed under addition?
Natural (ℕ), integer (ℤ), rational (ℚ), real (ℝ) and complex (ℂ) numbers are all closed under addition.
Is the set of natural numbers closed under addition?
Yes, when you add any group of natural numbers, the sum will also be a natural number.
When are complex numbers closed under addition?
Quite simply, they are closed under addition. No "when".
