This is from a textbook on topology:
A subset W of the set Z of integers is said to be closed under addition if given any elements w and w′ of W, w+w′∈W.
Prove that there is a maximal subset of Z which is closed under addition and does not contain 9
My problem is as follows: The set of all multiples of 4 does not contain 9 and is closed under addition, the same holds for all multiples of 5. A maximal set must contain both sets, but if that set is closed under addition, we get that 9 is a member of the set.
EDIT: Thanks for the clarification, on the definition I was really lost. Can anyone give me a sketch of a proof?