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Midterm

# Midterm + solution of Winter 2005 useful for practice

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University of Waterloo

Computer Science

CS 245

Nancy Day

Fall

Description

University of Waterloo
Midterm Examination
SOLUTION SET
Term: Winter Year: 2005
Student Name
UW Student ID Number
Course Abbreviation and Number CS 245
Course Title Logic and Computation
Sections SE112 - 001 and CS245 - 001, 002, 003
Instructor Shalini Aggarwal, Nancy Day
Date of Exam Thursday, February 10, 2005
Time Period Start time: 4:30 p.m. End time: 6:30 p.m.
Duration of Exam 2 hours
Number of Exam Pages 12 pages (including this cover sheet)
Exam Type Closed book
Additional Materials Allowed NO ADDITIONAL MATERIALS ALLOWED
• Write your name and student number at the bottom of every page.
• Write all solutions on the exam. The booklets are for scratch work.
• There are blank truth tables on the last page for use with any question.
• Good luck everyone!
Question Mark Max Marker Question Mark Max Marker
1 5 7 9
2 5 8 6
3 7 9 9
4 12 10 14
5 9 11 11
6 13
Total 100
Name UW Student ID (page 1 of 12) 1 (5 Marks) Short Answer
1. Can transformational proof be used to show an argument in propositional logic is valid?
If so, what do you prove in transformational proof about an argument of the form p ⊢ q
where p and q may be compound formulas?
If not, explain why.
Yes, transformational proof can be used to show an argument is valid. You would prove
p ⇒ q ⇚⇛ true.
2. If ¬a is a contingent formula in propositional logic, which of the following describes a?
satisﬁable, contingent
3. What is the name of the argument forms identiﬁed by Aristotle?
Syllogisms
4. What is the problem with using a proof procedure that is not sound?
If a proof procedure is not sound then it might be possible to prove an argument is valid when
it is not.
5. What must be true about a set of propositional logic formulas in order to be able to use
natural deduction to prove the set is consistent?
We can use natural deduction to show a set of formulas is consistent if and only if the con-
junction of the formulas is a tautology.
Name UW Student ID (page 2 of 12) 2 (5 Marks) Propositional Logic: Formalization
Formalize the following sentences in propositional logic. Show the phrase associated with each
prime proposition.
1. Michael Jackson is a software engineer only if pigs ﬂy.
j ⇒ p
where
• j is “Michael Jackson is a software engineer”
• p is “Pigs ﬂy”
2. I will go sledding exactly if school is cancelled and it is not windy.
s ⇔ c ∧ ¬w
where
• s is “I will go sledding”
• c is “School is cancelled”
• w is “It is windy”
Name UW Student ID (page 3 of 12) 3 (7 Marks) Propositional Logic: Boolean Valuations
Provide a Boolean valuation in which the following two formulas have diﬀerent truth values:
¬(a ∨ b ∨ c) ¬a ∧ b ⇔ a ∨ c
Demonstrate that the formulas have diﬀerent truth values in this Boolean valuation. Show your
work. You may refer to the truth tables on the last page.
A Boolean valuation in which the formulas have diﬀerent truth values is:
v(a) = F, v(b) = T, v(c) = T
v(¬(a ∨ b ∨ c))
= NOT (v(a) OR v(b) OR v(c))
= F
v(¬a ∧ b ⇔ a ∨ c)
= (NOT v(a) AND v(b)) IFF (v(a) OR v(c))
= T
Name UW Student ID (page 4 of 12) 4 (12 Marks) Propositional Logic: Transformational Proof
Prove the following using transformational proof:
(c ⇒ ¬a ⇒ b) ∧ (b ⇔ ¬a) ∧ ¬(b ∧ ⇚⇛ (a ∨ b) ∧ (¬b ∨ ¬a)
(c ⇒ ¬a ⇒ b) ∧ (b ⇔ ¬a) ∧ ¬(b ∧ a)
⇚⇛ (¬c ∨ (¬a ⇒ b)) ∧ (b ⇔ ¬a) ∧ ¬(b ∧ a) Impl
⇚⇛ (¬c ∨ (¬a ⇒ b)) ∧ (¬a ⇒ b) ∧ (b ⇒ ¬a) ∧ ¬(b ∧ a) Equiv
⇚⇛ (¬a ⇒ b) ∧ (b ⇒ ¬a) ∧ ¬(b ∧ a) Simp II
⇚⇛ (¬¬a ∨ b) ∧ (¬b ∨ ¬a) ∧ ¬(b ∧ a) Impl × 2
⇚⇛ (a ∨ b) ∧ (¬b ∨ ¬a) ∧ ¬(b ∧ a) Neg
⇚⇛ (a ∨ b) ∧ (¬b ∨ ¬a) ∧ (¬b ∨ ¬a) DM
⇚⇛ (a ∨ b) ∧ (¬b ∨ ¬a) Idemp
Name UW Student ID (page 5 of 12) 5 (9 Marks) Propositional Logic: Semantic Tableaux
Prove the following is a valid argument using semantic tableaux. Do not use any logical laws from
transformational proof in your semantic tableaux proof.
¬a ⇒ b ∨ c, b ⇒ d, ¬(c ∨ d) |ST a
1 ¬a ⇒ b ∨ c
2 b ⇒ d
3 ¬(c ∨ d)
4 ¬a
NOT-OR 3
5 ¬c
6 ¬d
IMPLIES 2
8 d

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