8. 1. Predicate Logic

Try these first

1. 1.

   Define what a predicate is.

   Hint

   Think about how statements relate to variables.

2. 2.

   What does the universal quantifier '∀x' signify?

   Hint

   Consider the meaning of 'for all' in logic.

3. 3.

   What does the symbol '∀' represent in predicate logic?

   • A. There exists
   • B. For all
   • C. Not necessarily

   Hint

   Consider how we express 'all' in logical statements.

4. 4.

   True or False: Existential quantification asserts that all elements are true for a predicate.

   • True
   • False

   Hint

   Think about the requirement for 'exist' versus 'all'.

5. 5.

   Prove that '∀x (P(x) ∧ Q(x))' is logically equivalent to '∀x P(x) ∧ ∀x Q(x)'.

   Hint

   Use methods of logical equivalences and quantify simultaneously.

6. 6.

   In a domain of natural numbers, evaluate the truth of '∃x (P(x)) → ∀x (Q(x))'. What implications does this carry?

   Hint

   Consider the implications of assuming P for all x and how it connects to the consequent.