Properties of real numbers applied to subsets

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Question from Mark, a student:

Hello -
The questions that I have for you is do the properties of real numbers (such as the associative, commutative, identity, inverse, and distributive law) apply to ALL the subsets of real numbers? In other words, do all those properties work for the Natural Numbers? The Whole Numbers? And so on and so forth. I understand that they are all real numbers, but for instance: the identity is whenever you add zero to a number, you get that number back. But does that work with, say, with only the odd numbers? Zero isn't odd so can that property actually apply to JUST the odd numbers? Any consideration would be greatly appreciated!

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!
RD

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