THE ALGEBRA OF SETS: The following statements are basic consequences of the above definitions, with A, B, C, ... representing subsets of a universal set U.
1. A ∪ B = B ∪ A. (Union is commutative)
2. A ∩ B = B ∩ A. (Intersection is commutative)
3. (A ∪ B) ∪ C = A ∪ (B ∪ C). (Union is associative)
4. (A ∩ B) ∩ C = A ∩ (B ∩ C). (Intersection is associative)
5. A ∪ Φ = A.
6. A ∩ Φ = Φ.
7. A ∪ U = U.
8. A ∩ U = A.
9. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C).(Union distributes over intersection)
10. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C). (Intersection distributes over union)
11. A ∪ AC = U.
12. A ∩ AC = Φ.
13. (A ∪ B)C = AC ∩ BC. (de’ Morgan’s law)
14. (A ∩ B)C = AC ∪ BC. (de’ Morgan’s law)
15. A ∪ A = A ∩ A = A.
16. (AC)C = A.
17. A – B = A ∩ BC.
18. (A – B) – C = A – (B ∪ C).
19. If A ∩ B = Φ, then (A ∪ B) – B = A.
20. A – (B ∪ C) = (A – B) ∩ (A – C).
This algebra of sets is an example of a Boolean algebra, named after the 19th-century British mathematician George Boole, who applied the algebra to logic. The subject later found applications in electronics.
