Natural numbers are actually closed under addition. If you add any two if them, the result will always be another natural number.
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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.
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Addition.
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Addition and multiplication.
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Yes, when you add any group of natural numbers, the sum will
also be a natural number.