9.5.1. Existential Quantification Example

Try these first

1. 1.

   What does S(x) represent in the context of the given examples?

   Hint

   Think about what the statement is inferring about students.

2. 2.

   What is the difference between 'for all x, S(x) ∧ C(x)' and 'for all x, S(x) → C(x)'?

   Hint

   Focus on the implications of each statement.

3. 3.

   What is the correct representation of 'Every student in CS201 has studied calculus'?

   • ∀x (S(x) ∧ C(x))
   • ∀x (S(x) → C(x))
   • ∃x (S(x) ∧ C(x))

   Hint

   Focus on the implications.

4. 4.

   Is the statement 'There exists some x such that S(x) ∧ C(x)' true if no students are in CS201?

   • True
   • False

   Hint

   Revisit the meaning of 'some student'.

5. 5.

   Given a university with 100 students, construct a scenario demonstrating the difference between 'some students are taking CS201' and 'all students are taking CS201' using predicates.

   Hint

   Assess the total students in CS201 and how they relate to the predicates defined.

6. 6.

   Formulate a logical expression for 'No students in CS201 have failed calculus' and discuss how it can be represented differently.

   Hint

   Explore negation in existential quantification and its conversion to universal quantification.