8. 1.7. Logical Equivalence in Predicate World

Try these first

1. 1.

   Define a predicate function based on the statement 'x is an even number'.

   Hint

   Think about how you would express the property of being even.

2. 2.

   What does the notation ∀x P(x) signify?

   Hint

   Recall our discussion on universal quantification.

3. 3.

   What does '∀x P(x)' indicate in predicate logic?

   • A property holds true for at least one x
   • A property holds true for all x
   • A specific case where P(x) is true

   Hint

   Remember the meaning of ∀.

4. 4.

   True or False: An existential quantifier asserts something is true for every element in a domain.

   • True
   • False

   Hint

   Think about how many elements need to meet the property.

5. 5.

   Prove that if ∀x P(x) is true, then ∃x P(x) cannot be false.

   Hint

   Consider the implications of universal truth

6. 6.

   Construct a pair of predicates where logical equivalence does not hold in a specified domain.

   Hint

   Check the outcomes across different numbers.