SOLUTIONS FOR MATH 55, HOMEWORK #1
SOLUTIONS FOR MATH 55, HOMEWORK #1
Contact. If you see any errors in this solution set, or if you have any questions, feel free to email the GSI at isammis@math.berkeley.edu.
Note about problem set 1. Because this is the first problem set and because it’s the introduction to formal logic, many questions on this set are of the form “write down an x equivalent to y.” There’s no intermediate work in such a problem—one simply writes the answer down and moves on. This is generally not the case for this course; most problems will ask you to present an argument. On such problems a bare, unsupported assertion of an answer won’t suffice.
Problem 1.1.8. Let p, q, and r be the propositions p : You have the flu. q : You miss the final examination. r : You pass the course.
Write these propositions using p and q and logical connectives.
(a) p → q
(b) ¬q ⇐⇒ r
(c) q → ¬r
(d) p ∨ q ∨ r
(e) (p → ¬r) ∨ (q → ¬r)
(f ) (p ∧ q) ∨ (¬q ∧ r)
Solution. Because this is part of the chapter 1.1, we don’t have logical equivalences yet. That’s a shame—many of these solutions could be made to sound more
“natural” by first altering the form of the logic a bit.
(a) If you have the flu, you miss the final examination.
(b) You pass the course if and only if you take the final exam.
(c) If you miss the final examination, you will not pass the course.
(d) You either have the flu, miss the final examination, or pass the course (or any combination).
(e) If you have the flu, you will fail the course, or if you miss the final exam, you will fail the course. (Note that if we had chapter 1.2 at this point, we could simplify the statement first:
(p → ¬r) ∨ (q → ¬r) ⇐⇒ (¬p ∨ ¬r) ∨ (¬q ∨ ¬r)
⇐⇒ (¬p ∨ ¬q) ∨ ¬r
⇐⇒ ¬(p ∧ q) ∨ ¬r
⇐⇒ (p ∧ q) → ¬r
Typeset by AMS-TEX
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SOLUTIONS FOR MATH 55, HOMEWORK #1
write the more natural sentence “you will fail the course if you have a flu and miss the final examination.”
(f) Either you have the flu and miss the final exam, or you don’t miss the
Contact. If you see any errors in this solution set, or if you have any questions, feel free to email the GSI at isammis@math.berkeley.edu.
Note about problem set 1. Because this is the first problem set and because it’s the introduction to formal logic, many questions on this set are of the form “write down an x equivalent to y.” There’s no intermediate work in such a problem—one simply writes the answer down and moves on. This is generally not the case for this course; most problems will ask you to present an argument. On such problems a bare, unsupported assertion of an answer won’t suffice.
Problem 1.1.8. Let p, q, and r be the propositions p : You have the flu. q : You miss the final examination. r : You pass the course.
Write these propositions using p and q and logical connectives.
(a) p → q
(b) ¬q ⇐⇒ r
(c) q → ¬r
(d) p ∨ q ∨ r
(e) (p → ¬r) ∨ (q → ¬r)
(f ) (p ∧ q) ∨ (¬q ∧ r)
Solution. Because this is part of the chapter 1.1, we don’t have logical equivalences yet. That’s a shame—many of these solutions could be made to sound more
“natural” by first altering the form of the logic a bit.
(a) If you have the flu, you miss the final examination.
(b) You pass the course if and only if you take the final exam.
(c) If you miss the final examination, you will not pass the course.
(d) You either have the flu, miss the final examination, or pass the course (or any combination).
(e) If you have the flu, you will fail the course, or if you miss the final exam, you will fail the course. (Note that if we had chapter 1.2 at this point, we could simplify the statement first:
(p → ¬r) ∨ (q → ¬r) ⇐⇒ (¬p ∨ ¬r) ∨ (¬q ∨ ¬r)
⇐⇒ (¬p ∨ ¬q) ∨ ¬r
⇐⇒ ¬(p ∧ q) ∨ ¬r
⇐⇒ (p ∧ q) → ¬r
Typeset by AMS-TEX
1
2
SOLUTIONS FOR MATH 55, HOMEWORK #1
write the more natural sentence “you will fail the course if you have a flu and miss the final examination.”
(f) Either you have the flu and miss the final exam, or you don’t miss the