2.1. Logical Equivalence

Try these first

1. 1.

   Define what a tautology is and give an example.

   Hint

   Think of statements that cannot be false.

2. 2.

   What does the statement 'p if and only if q' represent?

   Hint

   Consider when both statements hold the same truth values.

3. 3.

   What is the definition of logical equivalence?

   • A condition where two propositions always yield the same truth value.
   • A statement that is occasionally true.
   • Two propositions can never be false.

   Hint

   Think of the relationship between truth values.

4. 4.

   True or False: A contradiction is a statement that is always true.

   • True
   • False

   Hint

   Consider its definition.

5. 5.

   Prove that (p → q) and ¬(p ∧ ¬q) are logically equivalent using any logical identities.

   Hint

   Start with rewriting one statement and use identities.

6. 6.

   Use truth tables to establish the equivalence of (p ∨ q) and ¬(¬p ∧ ¬q).

   Hint

   Outline all possibilities for p and q.