Given the following predicates:

C(x): x is an employee of company XYZ

D(x): x is paid for overtime

M(x): x is not a manager

Translate the following logical expression into English:

x (C(x) (D(x) M(x)))

Use De Morgan's Law to find the negation of each sentence:

(Hint:x (P(x) Q(x) is an acceptable final answer, but x (P(x) v Q(x)) is not)

a) x (P(x) Q(x))

b) x (P(x) Q(x))

Given the following Predicates where the domain of discourse is :

P(x): x is prime

R(x, y, z): x² + y² = z²

Find the propositions and give the truth value:

a) P(2)

b) R(4, 5, 6)

c) R(0.3, 0.4, 0.5)