4.11

Convert the following using predicate calculus that contain double quantifiers:

a. Any even integer is equal to twice some other even integer

b. Some natural number is no bigger than every natural number

4.14

Write the negation in predicate math:

a. All managers work for at least 40 hours a week.

- Let P(x): x is a manager, Q(x): x works for at least 40 hrs week

- Vx(P(x)-> Q(x))

b. Some Dolphins do not have gills

- Let P(x): x is dolphins, Q(x): x have gills

- x(P(x) ~Q(x))

c. Every rational number multiplied by 10 is positive

- Let P(x): x is a rational number, Q(x): x multiplied by 2 is positive

- Vx( P(x)-> Q(x))