# The Algebra Of Sets

**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 ∪ A^{C} = U.

12. A ∩ A^{C} = Φ.

13. (A ∪ B)^{C} = A^{C} ∩ B^{C}. (de’ Morgan’s law)

14. (A ∩ B)^{C} = A^{C} ∪ B^{C}. (de’ Morgan’s law)

15. A ∪ A = A ∩ A = A.

16. (A^{C})^{C} = A.

17. A – B = A ∩ B^{C}.

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.