7.3.1. Satisfiability of a Compound Proposition

Try these first

1. 1.

   Rewrite the implication 'p → q' using conjunction and negation.

   Hint

   Think about how implications can be transformed into disjunctions.

2. 2.

   Is the proposition 'True OR False' satisfiable? Why?

   Hint

   Remember, any disjunction is satisfied if at least one part is True.

3. 3.

   What does it mean for a proposition to be satisfiable?

   • It is always false
   • It can be made true with some assignments
   • It is a tautology

   Hint

   Think about the conditions that define satisfiability.

4. 4.

   True or False: A compound proposition can be satisfiable and a tautology at the same time.

   • True
   • False

   Hint

   Reflect on the definitions of tautologies versus satisfiability.

5. 5.

   Given the proposition '(p AND q) OR (NOT r)', find a truth assignment that satisfies it.

   Hint

   Look to satisfy any part of the disjunction.

6. 6.

   Formulate a proof showing that having just the negation operator can still create a valid logical expression.

   Hint

   Think about how negation transforms the logical landscape.