9.5. Further Examples

Try these first

1. 1.

   Translate 'All birds are animals' into predicate logic.

   Hint

   Think about how to represent 'all' in logic.

2. 2.

   What does '∃x (S(x) ∧ C(x))' represent?

   Hint

   Focus on the meaning of 'some'.

3. 3.

   What does the expression ∀x (S(x) → C(x)) imply?

   • Every student studies calculus
   • If a student is in CS201
   • then they studied calculus
   • Some students do not study calculus

   Hint

   Refer to the definition of implication.

4. 4.

   True or False: ∃y (P(y) ∧ ¬Q(y)) is true if there exists a number y that makes P(y) true and Q(y) false.

   • True
   • False

   Hint

   Think about the requirements of existential quantification.

5. 5.

   Given the expressions ∀x P(x) and ∃y (P(y) → Q(y)), analyze the conditions under which Q must hold true.

   Hint

   Use the relationships between universal and existential quantifiers.

6. 6.

   Design a scenario illustrating both universal and existential quantifiers in a classroom setting.

   Hint

   Translate the scenario into logical terms.