8. Use De Morgan's laws and other logical equivalences to negate the proposition p ∧ (p → q). Simplify your answer as much as possible so that the negation symbol only appears before the composite propositions p or q.
9. Give the converse, inverse, and contrapositive of the conditional statement, "If you enter the correct username and password, then you can access the site."
10. For the given conditional statement, determine its truth value and the truth values of its converse, inverse, and contrapositive.
"If x and y are odd integers, then x+y is even."
11. Reformulate the following statement so that it of the form "if p then q." Then, write its contrapositive.
"The square root of an irrational number is irrational."
12. Let the domain be all animals, and let P(x):= ‐x is a turtle‑ and Q(x):= ‐x is a reptile.‑
Determine the truth value of each of the following:
a. ∃x Q(x)
b. ∀x P(x)
c. ∀x(P(x) → Q(x))
d. ∃x P(x) ∧ ∀x Q(x)
13. Determine whether the following quantified propositions are true or false. Assume that the domain is nonnegative real numbers.
a. ∃x(x^2 = -1)
b. ∃x∀y(xy = 0)
c. ∀x∃y(x^2 + y^2 = 4)
d. ∀x∀y((x+y)/2 ≥ √xy)
14. Rewrite the following statements using quantifiers and propositional functions. Be sure to specify the domain. Then write the negation of each statement.
a. Every positive integer has a prime factorization.
b. Some infectious human diseases are not zoonotic.
c. There is no honest politician.
15. Write the negation of each of the quantified propositions from problem 13.