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List of Identities

 

  1. P  (P  P) idempotence of 
  2. P  (P  P) idempotence of 
  3. (P  Q)  (Q  P) commutativity of 
  4. (P  Q)  (Q  P) commutativity of 
  5. [(P  Q)  R]  [ P  (Q  R)] associativity of 
  6. [(P  Q)  R]  [ P  (Q  R)] associativity of 
  7. ¬ (P  Q)  [¬ P  ¬ Q] De-Morgan’s law
  8. ¬ (P  Q)  [¬ P  ¬ Q] De-Morgan’s law
  9. [P  (Q  R]  [( P  Q)  (P  R)] distributivity of  over 
  10. [P  (Q  R]  [( P  Q)  (P  R)] distributivity of  over 
  11. (P  True)  True
  12. (P  False)  False
  13. (P  False)  P
  14. (P  True)  P
  15. (P  ¬ P)  True
  16. (P  ¬ P)  False
  17. P  ¬ (¬ P) double negation
  18. (P  Q)  (¬ P  Q) implication
  19. (P  Q)  (P  Q)  (Q  P) equivalence
  20. [(P  Q)  R]  [P  (Q  R)] exportation
  21. [(P  Q)  [P  ¬ Q)  ¬ P absurdity
  22. [P  Q  (¬ Q  ¬ P) contrapositive
  1. Two formulae A and A* are said to be duals of each other if either one can be obtained from the other by replacing  by  and  by .
    If the formula A contains special variables 1 or 0, then its dual A* is obtained by replacing 1 by 0 and 0 by 1.
    e.g.,
    (i) Dual of (p
      q)  r is (p  q)  r
    (ii) Dual of (p  q)  0 is (p  q)  1.
  2. Tautology implications : A statement A is said to tautologically imply a statement B if and only if A  B is a tautology which is read as “A implies B”.
    The implications listed below have important applications which can be proved by truth tables :
    1. p ∧ q ⇒ p p ⇒ p ∨ q
    2. ~ p ⇒ p → q q ⇒ p → q
    3. ~ (p → q) ⇒ p ~ (p → q) ⇒ ~ q
    4. P ∧ (P → q) ⇒ q ~ p ∧ (p ∨ q) ⇒ q
    5. (p → q)  (q  r)  p  r





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