9. Nested Quantifiers

Try these first

1. 1.

   Translate 'Every student in the class has a book.' into logical notation.

   Hint

   Think about what 'every' implies in logic.

2. 2.

   What does ∃y: P(y) mean?

   Hint

   Look at the meaning of existential quantification.

3. 3.

   What does the statement '∀x: P(x) → Q(x)' imply?

   • P is true for some x
   • For all x
   • if P is true then Q is true
   • Q is true for all x

   Hint

   Focus on the relationship described by the statement.

4. 4.

   True or False: The expression ∃y ∀x: M(x,y) means there is one mother for every individual.

   • True
   • False

   Hint

   Think about how the order of quantifiers changes meaning.

5. 5.

   Design a case involving nested quantifiers that represents a family structure and describe the implications.

   Hint

   Focus on how individuals relate within the family.

6. 6.

   Create an argument using universal and existential quantifiers to prove a theorem about friendship.

   Hint

   Specify how friendship applies universally while existing uniquely.