Closure of Integers?

Question

Fellow Mathers, is the fact that the Integers are closed under addition and multiplication...

∀ a,b ∈ Z , a + b = c where c ∈ Z

∀ a,b ∈ Z , ab = c where c ∈ Z

...universal axioms, theorems, dependent upon the axioms your using, etc?

Thank you. - New Math Guy

Answer 1

It depends sensitively on your definitions. Assuming a given definition of $\mathbb{N}$, here is a perspective from which both are theorems. Let $\sim$ be an equivalence relation on the set $\mathbb{N} \times \mathbb{N}$, defined by: $$(a,b) \sim (c,d) \iff a+d = c+b.$$ Then, for each $(a,b) \in \mathbb{N}^{2}$, we can define the equivalence class, $$E_{a,b} = \{(c,d) \text{ such that } (c,d) \sim (a,b)\}.$$ It is not hard to see that the set of all such equivalence classes consists of:

• $E_{a+1,1}$ for $a \in \mathbb{N}$ (the so-called positive integers),
• $E_{1, a+1}$ for $a \in \mathbb{N}$ (the so-called negative integers), and
• $E_{1,1}$ ("zero")

and no other elements. We can then define $\mathbb{Z}$ to be the set of all these equivalence classes. You can define $$E_{a,b} + E_{c,d} = E_{a+c, b+d}$$ well now it's totally totally obvious that this is closed, and the other properties of addition are theorems.

On the other hand, you might can imagine giving someone only "part of the definition", such as telling them that $+$ is commutative, and telling them how to add positive numbers to integers (leaving the rest to be worked out). Then the fact that this uniquely defines a binary operation on $Z$ is a theorem, and some of the other properties are definitions.

Answer 2

They are not theorems but axioms of the integers. In other words, the closure laws for addition and multiplication are among the fundamental statements about integers and real numbers that we assume to be true.