1. Construct the logical matrices for the following propositional formulas:
• ( ¬ (p ∨ q) → (p ∧ q))
• (p → q) → ( ¬ (p ∨ q) → (q → p) )
  


2. Determine all other logical functors using Sheffer's disjunction |.
   (def.:1|1= 0, 1|0 = 0|1= 0|0 = 1)


3. Analyse the following propositions and answer, if John is or is not a rock singer.
• John is a teacher.
• It is not true that John is a teacher and John is rich.
• If John is  a rock singer then John is rich.
  


4. Which of the rules of inference is applied in the following reasoning:
   "If you know that you are dead, then you have died; If you know that you are dead, then you may not have died.
   Therefore, you do not know if you are dead".


5. Check, which of the following formulas is a tautology of the propositional calculus (apply the zero-one method and the apagogic method)
• ((p ∧ q) → r) → ((p → (q → r)),
• a → (b ∨c)) → ((a → b) ∨ (a →c)).
  


6. Point out the laws of the propositional calculus and the rules of inference applied in the proof of the theorem: for any sets A, B, C, holds A ∩ (B\C) = (A ∩ B)\C.
   


7. Write down the proposition which is true if and only if exactly one of the three sentences p, q, r is true.
   


8. Find the shortest form for the following instructions:
   
   • if p then I else if q then if not p then I else II fi else II fi fi
   • if ( p or q) then I else if  not p then  II else III fi fi
      
     
9. Are the following instructions equivalent?
   'if α then P else Q fi;'    and    'if α then P fi; if not α then Q fi;'
   
   
10. Is the following reasoning true: if (α → β) is a tautology and the instruction 'while α do P od' does not loop for ever, then 'while β do P od' does not loop for ever too.

 

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