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

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

Exercise 1.(50 pts) Translate the following sentences to
symbols, write a useful negation, and translate back to English the
negation obtained.
1. No right triangle is isoceles
2. For every positive real number x, there is a unique real
number y such that 2^y=x
.Exercise 2.(50 pts) Which of the following are true? Explain
your answer.
1.(∀x∈R)(x^2+ 6x+ 5 ≥ 0).
2.(∃x∈N)(x^2−x+ 41 is prime)

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

write the following sentences as quantified logical statements, using the universal and existential quantifiers, and defining predicates as needed.
Second, write the negations of each of these statements in the same way.
Finally, choose one of these statements to prove. If it is true, prove it, and if it is false, prove its negation. Your proof need not use symbols, but can be a simple explanation in plain English.
1. If m and n are positive integers and mn is...

Determine what is wrong with the following “proof” by induction
that all dogs are the same breed. Theorem 1. All dogs are the same
breed. Proof. For each natural number, let P(n) be the statement
“any set of n dogs consists entirely of dogs of the same breed.” We
demonstrate that for each natural number n, P(n) is true. P(1) is
true, because a set with only one dog consists entirely of dogs of
the same breed. Now, let k...

Use symbols to write the logical form of each of the following
arguments. Then state whether or not the argument is valid. If it
is valid, state which of the following rules of inference apply
(Modus Ponens - Method of Affirming, Modus Tollens - Method of
Denying, Generalization, Specialization, Elimination, Transitivity,
or Division by Cases). If the argument is not valid, state whether
the Inverse error or Converse error was made.
a) if n is an integer, then n is...

Consider the following (true) statement:
“All birds have wings but some birds cannot fly.”
Part 1
Write this statement symbolically as a conjunction of two
sub-statements, one of which is a conditional and the other is the
negation of a conditional.
Use three components (p, q, and r)
and explicitly state what these components correspond to in the
original statement.
Hint: Any statement in the form "some X cannot Y"
can be rewritten equivalently as “not all X can Y,”...

Which of the following statements has truth value?
(a) x + 2 = 5
(b) The one millionth digit of π is 4
(c) 3 * 7 = 37
(d) x is an even number.

You’re the grader. To each “Proof”, assign one of the following
grades:
• A (correct), if the claim and proof are correct, even if the
proof is not the simplest, or the proof you would have given.
• C (partially correct), if the claim is correct and the proof
is largely a correct claim, but contains one or two incorrect
statements or justications.
• F (failure), if the claim is incorrect, the main idea of the
proof is incorrect, or...

Let D = E = {−2, 0, 2, 3}. Write negations for each of the
following statements and determine which is true, the given
statement or its negation. Explain your answer
(i) ∃x ∈ D such that ∀y ∈ E, x + y = y.
(ii) ∀x ∈ D, ∃y ∈ E such that xy ≥ y.

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