Semiring of Sets: Examples

From now on X denotes a set and S denotes a family of subsets of X. Now we state the propositions: (1) Let us consider sets X1, X2, a family S1 of subsets of X1, and a family S2 of subsets of X2. Then {a×b, where a is an element of S1, b is an element of S2 : a ∈ S1 and b ∈ S2} = {s, where s is a subset of X1 ×X2 : there exist sets a, b such that a ∈ S1 and b ∈ S2 and s = a × b}. Proof: {a × b, where a is an element of S1, b is an element of S2 : a ∈ S1 and b ∈ S2} ⊆ {s, where s is a subset of X1×X2 : there exist sets a, b such that a ∈ S1 and b ∈ S2 and s = a× b} by [6, (96)]. (2) Let us consider sets X1, X2, a non empty family S1 of subsets of X1, and a non empty family S2 of subsets of X2. Then {s, where s is a subset of X1 × X2 : there exist sets x1, x2 such that x1 ∈ S1 and x2 ∈ S2 and s = x1 × x2} = the set of all x1 × x2 where x1 is an element of S1, x2 is an element of S2. (3) Let us consider sets X1, X2, a family S1 of subsets of X1, and a family S2 of subsets of X2. Suppose