| Theorem 6.2.2 Set Identities
(All sets A, B, C are subsets of a universal set U.) |
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|---|---|
| 1. Commutative Laws: For all sets A and B, | |
| (a) A ∪ B = B ∪ A | (b) A ∩ B = B ∩ A |
| 2. Associative Laws: For all sets A, B, and C, | |
| (a) (A ∪ B) ∪ C = A ∪ (B ∪ C) | (b) (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| 3. Distributive Laws: For all sets A, B, and C, | |
| (a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) | (b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| 4. Identity Laws: For every set A, | |
| (a) A ∪ φ = A | (b) A ∩ U = A |
| 5. Complement Laws: For every set A, | |
| (a) A ∪ Ac = U | (b) A ∩ Ac = φ |
| 6. Double Complement Law: For every set A, | |
| (Ac)c = A | |
| 7. Idempotent Laws: For every set A, | |
| (a) A ∪ A = A | (b) A ∩ A = A |
| 8. Universal Bound Laws: For every set A, | |
| (a) A ∪ U = U | (b) A ∩ φ = φ |
| 9. De Morgan's Laws: For all sets A and B, | |
| (a) (A ∪ B)c = Ac ∩ Bc | (b) (A ∩ B)c = Ac ∪ Bc |
| 10. Absorption Laws: For all sets A and B, | |
| (a) A ∪ (A ∩ B) = A | (b) A ∩ (A ∪ B) =A |
| 11. Complements of U and φ | |
| (a) Uc = φ | (b) φc = U |
| 12. Set Difference Law: For all sets A and B, | |
| A - B = A ∩ Bc | |