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Laws Of Set Theory

  1. Commutative Law
    A  B = B  A ; A  B = B  A.
  2. Associative Law
    A  (B  C) = (A  B)  C
    A  (B  C) = (A  B)  C
  3. Distributive Law
    A  (B  C) = (A  B)  (A  C)
    A  (B  C) = (A  B)  (A  C)
  4. Complement Law
    A  A° = U ; A  Ac = φ.
  5. Identity Law
    A φ = A = φ A
    A U = A = U A.
  6. Absorption Law A (A  B) = A ; A (A B) = A
  7. De Morgan’s Law (A B)c = Ac  Bc ; (A B)c = Ac  Bc
  8. Involution Law (Ac)c = A.

Obs. Using the distributive law, we can extend the above result for three sets A, B, C

| A  B  C | = (A  B)  C |

= | A  B | + | C | – | (A  B)  C |

= | A | + | B | – | A  B | + | C | – | (A  C)  (B  C) |

= | A | + | B | + | C | – | A  B | – [ | A  C | + | B  C | – A  B  C | ]

hence follows the result

= | A | + | B | + | C | – | A  B | – | A  C | – | B  C | + | A  B  C|]





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