The power set ℘(E) of all subsets of a universal set E is closed under the binary operations ∪(union) and ∩(intersection), + (symmetric difference) and the unary operation ′ (complementation). The following are some of the properties, or laws, that hold for subsets A, B, and C of E:

1. (i) A∪(B ∪ C)=(A ∪ B)∪C, A + (B + C)=(A + B) + C and A∩(B ∩ C)=(A ∩ B)∩C, the associative properties.

2. (ii) A ∪ B = B ∪ A, A + B = B + A, and A ∩ B = B ∩ A, the commutative properties.

3. (iii) A∪Ø = A, A + Ø = A and A∩Ø= Ø, where Ø is the empty set.

4. (iv) A ∪ E = E and A ∩ E = A.

5. (v) A ∪ A = A, A + A = Ø and A ∩ A = A.

6. (vi) A∩(B ∪ C)=(A ∩ B)∪(A∩∩C), A + (B ∩ C)=(A + B)∩(A + C), and A∪(B ∩ C)=(A ∪ B)∩(A ∪ C), the distributive properties.

7. (vii) A ∪ A′ = E, A + A′ = E, and A ∩ A′=Ø.

8. (viii) E′=Ø and Ø′ = E.

9. (ix) (A′)′ = A.

10. (x) (A ∪ B)′ = A′∩B′ and (A ∩ B)′ = A′∪B′, De Morgan’s laws.

The application of these laws to subsets of E is known as the algebra of sets. Despite some similarities with the algebra of numbers, there are important and striking differences. If |E| = n then (℘(E), +, ∩) is isomorphic as a ring to (ℤ₂ⁿ, +, ×). See Boolean algebra.