Can these sentences be used to define a Boolean Algebra?

Question

T = {X ⊆ ℕ : X is finite} ∪ {X ⊆ ℕ : ℕ\X is finite}

x∧y = x∩y

x∨y = x∪y

x' = ℕ\x

Zero = ∅

One = ℕ

I think it is correct but I am not sure if the finite (or infinite) set can make influence in defining Boolean Algebra.

Answer 1

In general, for any set $X$, the powerset $\mathcal{P}(X)$ together with the following operations is a Boolean algebra :

• $x\wedge y := x\cap y$
• $x\vee y := x\cup y$
• $x' := X \setminus x$
• $\top := X$
• $\bot := \emptyset$

Here, $T$ is just a subalgebra of $\mathcal{P}(\mathbb{N})$ :

• $\top, \bot \in T$
• $T$ is stable under $\wedge$, $\vee$ and $'$

Hence, $T$ is a Boolean algebra.