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Logical connectives (Posted on 2012-01-24) Difficulty: 3 of 5
Let P|Q mean "P and Q are both false." P|Q is true if P and Q are both false, and false otherwise. Define the following logical connectives just using P, Q, |, and parentheses.

1. ~P (not P)
2. P&Q (P and Q)
3. PVQ (P or Q)
4. P->Q (P implies Q)
5. P<->Q (P if and only if Q)

See The Solution Submitted by Math Man    
Rating: 4.5000 (2 votes)

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Solution NOR | Comment 5 of 6 |

   Truth table for | or NOR:
      P Q  P|Q
      T T   F
      T F   F
      F T   F
      F F   T
   ~ (not) in terms of | (NOR):
      P  ~P  (P|P)  
      T  F     F
      F  T     T
         ^     ^
   & (and) in terms of | (NOR):
      P Q  P&Q  (P|P)|(Q|Q)
      T T   T     F  T  F
      T F   F     F  F  T
      F T   F     T  F  F
      F F   F     T  F  T
            ^        ^
   v (or) in terms of | (NOR):
      P Q  PvQ  (P|Q)|(P|Q)
      T T   T     F  T  F
      T F   T     F  T  F
      F T   T     F  T  F
      F F   F     T  F  T
            ^        ^
   -> (implies) in terms of | (NOR):
      P Q  P->Q  ((P|P)| Q)|((P|P)| Q)
      T T   T       F  F T T      F
      T F   F       F  T F F      T
      F T   T       T  F T T      F
      F F   T       T  F F T      F
            ^              ^   
   <-> (iff) in terms of | (NOR):
      P Q  P<->Q  ((P|Q)| P)|((P|Q)| Q) 
      T T    T       F  F T T   F  F T
      T F    F       F  F T F   F  T F
      F T    F       F  T F F   F  F T
      F F    T       T  F F T   T  F F
             ^              ^
Note: NOR is commutative, but not associative.
Note: NOR and NAND are universal connectives -
      All of the sixteen logical connectives
      can be defined in terms of either.   

  Posted by Bractals on 2012-01-24 16:27:58
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