2.6.1. Summary of Concepts Introduced

Try these first

1. 1.

   What is a tautology? Give an example.

   Hint

   Think of statements that cover all truth values.

2. 2.

   Define a contradiction and give an example.

   Hint

   Look for examples where one part conflicts with another.

3. 3.

   Which of the following is always true?

   • p ∧ ¬p
   • p ∨ ¬p
   • p → ¬p

   Hint

   Consider what happens if p is true or false.

4. 4.

   Is p → q equivalent to ¬q → ¬p?

   • True
   • False

   Hint

   Think of the meaning of each side of the equivalence.

5. 5.

   Consider the statements p = 'It rains' and q = 'The ground is wet'. Show using logical equivalence that 'If it does not rain, then the ground is not wet' is equivalent to 'If the ground is wet, then it has rained'.

   Hint

   Start with rewriting the statements using known identities.

6. 6.

   Using De Morgan's Law, simplify ¬(p ∨ q).

   Hint

   Focus on how negation interacts with conjunctions and disjunctions.