Propositional Logic Rules

1. COMMUTATIVE
   • $P \wedge Q \Leftrightarrow Q \wedge P$
   • $P \vee Q \Leftrightarrow Q \vee P$
2. ASSOCIATIVE
   • $P \wedge (Q \wedge R) \Leftrightarrow (P \wedge Q) \wedge R$
   • $P \vee (Q \vee R) \Leftrightarrow (P \vee Q) \vee R$
3. DISTRIBUTIVE
   • $P \wedge (Q \vee R) \Leftrightarrow P \wedge Q \vee P \wedge R$
   • $P \vee (Q \wedge R) \Leftrightarrow (P \vee Q) \wedge (P \vee R)$
4. IDEMPOTENT (or Tautology)
   • $P \wedge P \Leftrightarrow P$
   • $P \vee P \Leftrightarrow P$
5. ABSORBTION
   • $P \wedge (P \vee Q) \Leftrightarrow P$
   • $P \vee (P \wedge Q) \Leftrightarrow P$
6. COMPLEMENTATION
   • $P \wedge \neg P \Leftrightarrow F$ (or 0)
   • $P \vee \neg P \Leftrightarrow T$ (or 1)
7. LAW OF INVOLUTION
   • (Double Complementation)
     $\neg \neg P \Leftrightarrow P$
8. LAWS OF DEMORGAN
   • $\neg P \wedge \neg Q \Leftrightarrow \neg (P \vee Q)$
   • $\neg (P \wedge Q) \Leftrightarrow \neg P \vee \neg Q$
9. IDENTITY ELEMENTS
   • Disjunction:
     • I $P \vee T \Leftrightarrow T$
     • II $P \vee F \Leftrightarrow P$
   • Conjunction:
     • I $P \wedge T \Leftrightarrow P$
     • II $P \wedge F \Leftrightarrow F$
10. EXCLUSIVE OR
    • $A \oplus B \Leftrightarrow A \wedge \neg B \vee \neg A \wedge B$
    • $\neg(A \oplus B) \Leftrightarrow A \wedge B \vee \neg A \wedge \neg B$
11. RELATION OF IMPLICATION
    • $P \Rightarrow Q \equiv \neg P \vee Q$
12. MODUS PONENS (Rule of Detachment)
    $P \Rightarrow Q$
    $P$
    $----$
    $Q$
13. MODUS TOLENS
    $P \Rightarrow Q$
    $\neg Q$
    $----$
    $\neg P$
    
14. RESOLUTION (Law of Syllogism)
    $P \vee Q$
    $\neg P \vee R$
    $----$
    $Q \vee R$
    
15. AND ELIMINATION
    $P \wedge Q$
    $----$
    $P$
    $Q$
    
16. DISJUNCTIVE SYLLOGISM
    $P$
    $----$
    $P \vee Q$
    
17. Basic Truth table:

    P  Q   V        P  Q  &
    -----------------------
    0  0   0        0  0  0
    0  1   1        0  1  0
    1  0   1        1  0  0
    1  1   1        1  1  1
    

18. Parallel and Series Circuits
    

         Parallel
    
              /
          ---   ---
         |    p    |
     A --           -- B               Also know: AND Gate
         |    /     |                             OR Gate
          ---   ---
              q
    
         Series
         
           /     /
     A ---   ---   --- B
           p     q