



 
Hi Mark: You've basically understood it perfectly  these properties restrict to any subset so long as any new element required by an operation or axiom is there. So the odd numbers are not closed under addition because there does not exist an odd number which is (say) 1+1. The natural numbers do not have additive inverses. To take your example, the odd numbers do not have the "additive zero property" because there does not exist an odd number "zero" such that it can be added to any other odd number without changing its value. The associative and commutative laws work in any subset if they are understood as saying (eg) "if a+b is defined, then b+a is defined and they are equal." Good Hunting!  


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