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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.
Let + (addition) be a binary operation on the set of odd numbers S. The set S is closed under + if for all a, b ϵ S, we also have a + b ϵ S. Let 3, 5 ϵ the set of odd numbers 3 + 5 = 8 (8 is not an odd number) Since 3 + 5 = 8 is not an element of the set of the odd numbers, the set of the odd numbers is not closed under addition.
Yes, because suppose that 'a' and 'b' are both arbitrary integers. Then (a-b) or (b-a) will then provide you with another integer. Suppose that the integer you are given from (a-b) is not unique. Then we have: (a-b)=c and (a-b)=c' Then, trivially, since (a-b)=(a-b), we have c=c'. Thus it is closed under subtraction.
You can't. Adding any two odd numbers always gives an even number, which is not a member of the set of odd numbers.
1 No. 2 No. 3 Yes.