Write all axioms and properties for the Boolean algebra of sets:
$S = set$
$(P(S), \cap$$, ∪ , complement; ∅, S)$
I know the axioms of Boolean algebra but I am not sure how to translate that to a power set
$(B, + ,·,' ; 0,1 )$
1) $+,·$ are closed operations in $B$:
$x + y ∈ B$
$x · y ∈ B, ∀ x, y ∈ B$
2) identity elements:
$x + 0 = 0 + x = x$
$x ·1 = 1·x = x, ∀ x ∈ B$
3) +,· are commutative:
$x + y = y + x$
$x·y = y·x, ∀ x, y ∈ B$
4) $+,·$ are distributive one relative to the other one:
$(x + y)·z = x·z + y·z$
$(x·y) + z = (x + z)(y + z), ∀ x, y, z ∈ B$
5) $∀ x ∈ B, ∃ x' ∈ B$, the complement of $x$, such that:
$x + x' = 1 and x x' = 0$
6) There are at least two elements in$ B$: $0 ≠ 1$