ID # 1226043119 - Written Homework Week #1
pdf
keyboard_arrow_up
School
Arizona State University *
*We aren’t endorsed by this school
Course
MAT 243
Subject
Philosophy
Date
Apr 3, 2024
Type
Pages
2
Uploaded by ChiefTank13657
ID # 1226043119 - Written Homework Week #1 1.
Is ”Johann Sebastian Bach is the greatest of all the Baroque composers” a proposition? Explain.
A.
This is not either true or false as it is a subjective statement which could be both true and false depending on subjective criteria. Thus it is not a proposition. 2.
(a) Write the negation of ”Hikaru is taller than Yutaka”. Your (verbally given) negation must not contain any words or phrases that explicitly express negation, such as ”not”, ”it is untrue that”, ”is false”, ”is incorrect”, etc.
A.
Hikaru is the same height as or shorter than Yutaka.
(b)
Write the fully simplified negation of 3 < x ≤ 4
. The variable x
denotes a real number. A.
x ≥ 3 OR x < 4 (c)
Use De Morgan to given an equivalent statement of ”It’s not true that Finley is rich and famous.”
A.
Finley is not rich or not famous. 3.
Is the conditional statement ”If a human being has 7 heads, then they have 11 arms” true or false? Explain.
A.
Based on human anatomy, a human being does not have 7 heads. So the statement is based on a false premise, and is thus false. 4.
Rephrase the following statements in standard ”if.. then” form:
(a)
”We are buying a new TV only if the old TV breaks down.”
A.
If we are buying a new TV
then the old TV broke down.
(b)
”In the United States, a good credit score is necessary for obtaining a loan.”
A.
In the United States, if you obtain a loan then you have a good credit score. (c)
Unless you make me a better offer, I will keep my current job.
A.
If you do not make me a better offer then I will keep my current job. (d)
The observation of faster than light travel would be sufficient reason to question relativity theory.
A.
If we observe faster than light travel then there would be reason to question relativity theory.
5.
Rephrase verbally in equivalent only if
, sufficient
, necessary
, contrapositive
and unless form: ”if we had an FTL drive, then we could visit the stars”. A.
Answers:
a.
We would have an FTL drive only if we could visit the stars. b.
Having an FTL drive is a sufficient condition for visiting the stars. c.
Being able to visit the stars is a necessary condition for having an FTL drive. d.
If we cannot visit the stars, then we do not have an FTL drive. e.
We could visit the stars unless we did not have an FTL drive. 6.
Rephrase in contrapositive form:
(a)
”If you are taller than 6 ft, then it is unpleasant for you to travel in economy class.” Your contrapositive must not contain explicit ref- erences to negation. Assume that the negation of ”unpleasant” is ”pleasant”.
A.
If it is pleasant for you to travel in economy class, then you are 6 ft or shorter. (b)
”If x ≥ 0 and y ≥ 0 then xy ≥ 0” where x, y are real numbers.
A.
If xy < 0 then x < 0 or y < 0. 7.
Give a logically equivalent expression to ¬p → q that does not use the conditional. Justify your answer.
A.
¬p → q ≡ ¬(¬p) ∨
q (Implication as Disjunction) ≡ p ∨
q (Double negation) 8.
Use logical equivalences to simplify (
p → q
) → (
¬p → ¬q
) until you have at most one occurrence of each variable p, q remaining. Identify all logical equivalences by name. You will not receive credit for a truth table solution.
A.
(
p → q
) → (
¬p → ¬q
)
≡ ¬( p → q
) ∨ (
¬p → ¬q
) (Implication as Disjunction) ≡ ¬( ¬p ∨ q
) ∨
(
¬(¬p) ∨ ¬q) (Implication as Disjunction) ≡ (p ∧ ¬q)
∨
(p ∨ ¬q) (De Morgan’s Law, Double negation) ≡ (p ∨ (p ∨ ¬
q))
∧
(
¬
q ∨ (p ∨ ¬
q))
(Distributive Law) ≡ ((p ∨ p) ∨ ¬
q)
∧
(
¬
q ∨ (
¬
q ∨ p
))
(Associative Law, Commutative Law) ≡ ((p ∨ p) ∨ ¬
q)
∧
((
¬
q ∨ ¬
q) ∨ p))
(Associative Law) ≡ (p
∨ ¬
q)
∧
(
¬
q ∨ p)
(Idempotent Law) ≡ (p
∨ ¬
q)
∧
(p
∨ ¬
q) (Commutative Law) ≡ p
∨ ¬
q (Idempotent law) Thus, (
p → q
) → (
¬p → ¬q
) ≡ p
∨ ¬
q
1
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help