indexing boolean algebra

Boolean algebra can be encoded by indexing, for example, a venn diagram for two sets: \(A\) and \(B\) divide diagram to four parts: \(1, 2, 3, 4\). Thus

1. \(A = \{1, 2\}\)
2. \(B = \{1, 3\}\)
3. \(A \cup B = \{1, 2, 3\}\)
4. \(A \cap B = \{1\}\)
5. \((A \cup B)' = \{4\}\)

The benefit of the encoding is the encoding let any sets can be treated as finite sets operation. So for any set check a boolean algebra is valid, indexing helps you check them by computer.

(define I (set 1 2 3 4))
(define A (set 1 2))
(define B (set 1 3))
(define ∅ (set))
(define ∩ set-intersect)
(define ∪ set-union)
(define (not S)
  (set-subtract I S))
(define (→ A B)
  (∪ (not A) B))
(define (- A B)
  (∩ A (not B)))
(define (≡ A B)
  (∪ (∩ A B) (∩ (not A) (not B))))

(equal? (→ A B) (not (- A B)))

(equal? (∩ A (not (- A B))) ∅)