Question

3. Transform each informal argument into a formalized wff. Then give a formal proof of the wff. (a) Every student likes cake and likes ice cream. Fred is a student. Therefore, some student likes cake and likes ice cream. (b) Every even number is divisible by 2. There is an even number. Therefore, there is a number which is divisible by 2.

Answer #1

Please explained using formal proofs in predicate logic
In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) ∀xP' (x) → ∃x[P(x) → Q' (x)].
(b) ∃x[P(x) → Q' (x)] → QxP' (x).

In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) ∀xA(x) → ∃x[B(x) → A(x)].
(b) ∃x[B(x) → A(x)] → ∃xA(x).

In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) [∀xA(x) ∨ ∀xB(x)] → ∀x[A(x) ∨ B(x)].
(b) [∃xA(x) ∧ ∃xB(x)] → ∃x[A(x) ∧ B(x)].

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...

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...

Is there a set A ⊆ R with the following property? In each case
give an example, or a rigorous proof that it does not exist.
d) Every real number is both a lower and an upper bound for
A.
(e) A is non-empty and 2inf(A) < a < 1 sup(A) for every a
∈ A.2
(f) A is non-empty and (inf(A),sup(A)) ⊆ [a+ 1,b− 1] for some
a,b ∈ A and n > 1000.

Consider the argument: "Each of five roommates, Melissa, Aaron,
Ralph, Veneesha, and Keeshawn, has taken a course in discrete
mathematics. Every student who has taken a course in discrete
mathematics can take a course in algorithms. Therefore, all five
roommates can take a course in algorithms next year."
Let r(x) denote "x is one of the five
roommates listed".
Let d(x) denote "x has taken a course
in discrete mathematics".
Let a(x) denote "x can take a course
in algorithms"....

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...

(1) Explain why the assumption of convex preferences
implies that “averages are preferred to extremes.” Make both a
formal argument and an intuitive one (that is, an explanation that
can be understood by the “man on the street.”)
(2) What does the negative slope of an indifference
curve imply about a consumer’s tastes for the two goods? How would
this change if one of the goods wasn’t a “good” at all (but instead
a “bad”…something people do not like)?
(3)...

A researcher is interested in whether the number of years of
formal education is related to a person's decision to never smoke,
continue to smoke, or quit smoking cigarettes. The data below
represent the smoking status by level of education for residents of
the United States 18 years or older from a random sample of 400
residents. Round all numeric answers to four decimal places.
Smoking Status
Education Level
Current
Former
Never
Less than high school
36
18
34
High...

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