Question

2. Which of the following is a negation for ¡°All dogs are
loyal¡±? More than one answer may be correct.
a. All dogs are disloyal. b. No dogs are loyal.
c. Some dogs are disloyal. d. Some dogs are loyal.
e. There is a disloyal animal that is not a dog.
f. There is a dog that is disloyal.
g. No animals that are not dogs are loyal.
h. Some animals that are not dogs are loyal.

3. Write a formal negation for each of the following statements:
a. * fish x, x has gills.
b. * computers c, c has a CPU.
c. ∃ a movie m such that m is over 6 hours long.
d. ∃ a band b such that b has won at least 10 Grammy
awards.

5. Write a negation for each of the following statements.
a. Any valid argument has a true conclusion.
b. Every real number is positive, negative, or zero.

10. * computer programs P, if P compiles without error messages,
then P is correct.

12. Statement: The product of any irrational number
and any rational number is irrational.Proposed negation: The product of any irrational number
and any rational number is rational.

15. Let D = {−48, −14, −8, 0, 1, 3, 16, 23, 26, 32, 36}.
Determine which of the following statements are true and
which are false. Provide counterexamples for those statements
that are false.
a. *x ∈ D, if x is odd then x > 0.
b. *x ∈ D, if x is less than 0 then x is even.
c. *x ∈ D, if x is even then x ≤ 0.
d. *x ∈ D, if the ones digit of x is 2, then the tens digit is
3 or 4.
e. *x ∈ D, if the ones digit of x is 6, then the tens digit is
1 or 2.

19. *n ∈ Z, if n is prime then n is odd or n = 2.

23. If a function is differentiable then it is continuous.

In 29, for each statement in the referenced exercise write the
converse, inverse, and contrapositive. Indicate as best as you can
which among the statement, its converse, its inverse, and its contrapositive
are true and which are false. Give a counterexample
for each that is false.

29. Exercise 19

40. Being divisible by 8 is a sufficient condition for being divisible
by 4.

42. Passing a comprehensive exam is a necessary condition for
obtaining a master’s degree.

14. Consider the following statement:
∃x ∈ R such that x^2 = 2.
Which of the following are equivalent ways of expressing
this statement?
a. The square of each real number is 2.
b. Some real numbers have square 2.
c. The number x has square 2, for some real number x.
d. If x is a real number, then x^2 = 2.
e. Some real number has square 2.
f. There is at least one real number whose square is 2.

Answer 1