4

• Laws of Algebra of Sets

  • Commutative Laws
  • Associative Laws
  • Distributive Laws
  • Identity Laws
  • Inverse or Complement Laws
  • Idempotent Laws
  • Involution Law
  • De Morgan's Law
  • Absorption Laws
  • Set Difference Law
• Commutative Laws


  𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴

  𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴
• Associative Laws


  𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶

  𝐴 ∩ (𝐵 ∩ 𝐶) = (𝐴 ∩ 𝐵) ∩ 𝐶
• Distributive Laws


  𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)

  𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)
• Identity Laws


  𝐴 ∪ Ø = 𝐴

  𝐴 ∩ 𝑈 = 𝐴

  𝐴 ∪ 𝑈 = 𝑈

  𝐴 ∩ Ø = Ø
• Inverse or Complement Laws


  𝐴 ∪ 𝐴𝑐 = 𝑈

  𝐴 ∩ 𝐴𝑐 = Ø

  𝑈𝑐 = Ø

  Ø𝑐 = 𝑈
• Idempotent Laws


  𝐴 ∪ 𝐴 = 𝐴

  𝐴 ∩ 𝐴 = 𝐴
• Involution Law


  (𝐴𝑐)𝑐 = 𝐴
• De Morgan's Law


  (𝐴 ∪ 𝐵)𝑐 = 𝐴𝑐 ∩ 𝐵𝑐

  (𝐴 ∩ 𝐵)𝑐 = 𝐴𝑐 ∪ 𝐵𝑐
• Absorption Laws


  𝐴 ∪ (𝐴 ∩ 𝐵) = 𝐴

  𝐴 ∩ (𝐴 ∪ 𝐵) = 𝐴
• Set Difference Law


  𝐴 - 𝐵 = 𝐴 ∩ 𝐵𝑐
• The aforementioned laws can be verified by the use of the Venn-Euler diagram
• Idempotent Law


  • 𝐴 ∪ 𝐴 = 𝐴

  𝐴 ∩ 𝐴 = 𝐴
• Associative Law


  • 𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶
• The principle of duality of sets states that when the operations of union and intersection, empty set and the universal set or any of the laws of sets are interchanged, a new valid equation is formed
• Principle of Duality


  • (𝐵 ∪ 𝐶) ∩ 𝐴 = (𝐵 ∩ 𝐴) ∪ (𝐶 ∩ 𝐴)

  (𝐵 ∩ 𝐶) ∪ 𝐴 = (𝐵 ∪ 𝐴) ∩ (𝐶 ∪ 𝐴)
• Proving Set Identities


  Example 1: (𝐴 ∪ 𝐵) ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐶) ∪ 𝐵

  Example 2: 𝐴𝑐 ∪ (𝐵 ∪ 𝐶)𝑐 = (𝐴 ∩ 𝐵)𝑐 ∩ (𝐴 ∩ 𝐶)𝑐