8. 1.3. Multi-valued Predicate Functions

Try these first

1. 1.

   What is a predicate function?

   Hint

   Think about how it describes elements.

2. 2.

   Define universal quantification.

   Hint

   It starts with a symbol that often resembles a loop.

3. 3.

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

   • P(x) holds for some x.
   • P(x) holds for all x.
   • P(x) is always false.

   Hint

   Consider how it encompasses all elements.

4. 4.

   True or False: Existential quantification asserts that a predicate is true for every element in a domain.

   • True
   • False

   Hint

   Think about ‘at least one’ versus ‘all’.

5. 5.

   Given the predicates P(x) = (x^2 > 0) and Q(x) = (x ≠ 0), prove that ∀x(P(x) → Q(x)).

   Hint

   Examine the conditions under which P will lead to Q. Just analyze both predicates together.

6. 6.

   Create an example where universal quantification fails in certain domains while holding true in others.

   Hint

   Try out edge cases when including boundary values.