# CS245 Study Guide - Midterm Guide: Proof Procedure, Propositional Calculus, Natural Deduction

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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 pand qmay 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 ¬ais 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

•jis “Michael Jackson is a software engineer”

•pis “Pigs ﬂy”

2. I will go sledding exactly if school is cancelled and it is not windy.

s⇔c∧ ¬w

where

•sis “I will go sledding”

•cis “School is cancelled”

•wis “It is windy”

Name UW Student ID (page 3 of 12)

## Document Summary

End time: 6:30 p. m: 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. Yes, transformational proof can be used to show an argument is valid. 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. Provide a boolean valuation in which the following two formulas have di erent truth values: Demonstrate that the formulas have di erent truth values in this boolean valuation. You may refer to the truth tables on the last page. A boolean valuation in which the formulas have di erent truth values is: v(a) = f, v(b) = t, v(c) = t v( (a b c))