1. The function of the rules of inference is to justify the steps of a proof. A) True B) False 2. A proof is a sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference. A) True B) False 3. Implication rules can be validly applied to part of a line. A) True B) False 4. According to the axiom of replacement, logically equivalent expressions may replace each other within the context of a proof. A) True B) False 5. Replacement rules are pairs of logically equivalent statement forms. A) True B) False 6. Conditional proof (CP) is a method that starts by assuming the negation of the required statement and then validly deriving a contradiction on a subsequent line. A) True B) False 7. Indirect proof (IP) is a method that starts by assuming the antecedent of a conditional statement on a separate line and then proceeds to validly derive the consequent on a separate line. A) True B) False 8. Invalid arguments cannot be proven as invalid using the methods of natural deduction. A) True B) False Document Preview:

Introduction to Formal Logic – Natural Deduction Examination – Spring 2014 Part I: True/False Instructions. Please circle ‘True ‘if the statement is true and ‘False’ if the statement is false. (2 pts each) 1. The function of the rules of inference is to justify the steps of a proof. A) TrueB) False 2. A proof is a sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference. A) TrueB) False 3. Implication rules can be validly applied to part of a line. A) TrueB) False 4. According to the axiom of replacement, logically equivalent expressions may replace each other within the context of a proof. A) TrueB) False 5. Replacement rules are pairs of logically equivalent statement forms. A) TrueB) False 6. Conditional proof (CP) is a method that starts by assuming the negation of the required statement and then validly deriving a contradiction on a subsequent line. A) TrueB) False 7. Indirect proof (IP) is a method that starts by assuming the antecedent of a conditional statement on a separate line and then proceeds to validly derive the consequent on a separate line. A) TrueB) False 8. Invalid arguments cannot be proven as invalid using the methods of natural deduction. A) TrueB) False 9. One of the rules of inference is DeMorgan’s theorem: ~(P v Q) :: (~P · ~ Q) A) TrueB) False 10. When an indirect proof is discharged, any lines derived from the assumption can no longer be used in the subsequent proof. A) TrueB) False Part II: Rules of Implication Instructions. Using the rules of implication, please provide the proofs for the following problems. Please provide justifications for each step. (7 pts each) 11. 1. T ? ( P v Q ) 2. S v ~ ( P v Q ) 3. ~S / ~ T 12. 1. ( S ? R ) · Q 2. R ? T / S ? T 13. 1. [ (…

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1. The Traveling Salesman problem (TSP) is a famous problem for which there is no known,

tractable solution (though efficient, approximate solutions exist). Given a list of cities and the

distances in between, the task is to find the shortest possible tour (a closed walk in which all

edges are distinct) that visits each city exactly once.

Consider the following algorithm for solving the TSP:

n = number of cities

m = n x n matrix of distances between cities

min = (infinity)

for all possible tours, do:

find the length of the tour

if length

min = length

store tour

a) What is the worst-case (big-O) complexity of this algorithm in terms of n (number of

cities)? You may assume that matrix lookup is O(1), and that the body of the ifstatement

is also O(1). You need not count the if-statement or the for-loop conditional

(i.e., testing to see when the for-loop is done), or any of the initializations at the start of

the algorithm. Clearly show the justification for your answer.

b) Given your complexity analysis, assume that the time required for the algorithm to run

when n = 10 is 1 second. Calculate the time required for n = 20 and show your work.

2. Suppose an algorithm solves a problem of size x in at most the number of steps listed in each

question below. Calculate the asymptotic time complexity (big-T or big-theta) for each example

below. Show your work, including values for c and x0along the way.

a) T(x) = 1

b) T(x) = 5x – 2

c) T(x) =3x^3+ 2 + x2

d) T(x) = log (x * 2x!)

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The value of F B. ds along closed path in a magnetic field is given by 6.28 X 10 T.m. What is the total current that passes througn this closed path?

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1. The structural model of an economy is given as below

C=400+0.8Yd Consumption function

I=350 Investment

G=450 Government expenditure

T=200 Tax

i) Derive the savings function (2 marks)

ii) Derive the government expenditure and tax multiplier (4 marks)

iii) Compute equilibrium income and consumption (4 marks)

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1) Title: initially this might be regarded as a working title, and ideally should mirror closely the content of the introduction. The title is particularly important as it will get people to stop and listen. 2) Introductory part: This informs the reader of the problem or situation and the context you are interested in. The goal is to give the viewers some basic academic information they can use to evaluate the significance of the research proposal. 3) Rationale for this study. – What is the research issue? – Why is it an issue? – Why is it an issue now? – What could this research shed light on? Document Preview:

1) Title: initially this might be regarded as a working title, and ideally should mirror closely the content of the introduction. The title is particularly important as it will get people to stop and listen. 2) Introductory part: This informs the reader of the problem or situation and the context you are interested in. The goal is to give the viewers some basic academic information they can use to evaluate the significance of the research proposal. 3) Rationale for this study. – What is the research issue?- Why is it an issue?- Why is it an issue now?- What could this research shed light on? You have to introduce the topic and clarify the significance of what you are trying to present. That is, you have to present research questions with a logical sequencing of facts. You have to give examples to illustrate your rationale. Be sure to have a conclusion that summarises your take-home message (why is it an issue now? what could this research shed light on?). Your introduction should be constructed so that it presents the desired information in a self- explanatory manner. You have to think of the introduction as a highly efficient, organised document upon which appear synopsis of the relevant information you want to convey (what is the research issue?), Just enough to get your point across. 4) Include references: You are required to identify previously conducted research in the area that you are focusing on. These sources need to be referred within your rationale section. Make it clear that you know what has been done in your area in the past and where your research will fit in. 5) Format: You have to make your introduction appealing. Simplify and include only relevant information. Be attentive to the structure and placement of your content. Write clearly. Make sure your introduction includes complete sentences and accurate spelling and punctuation. I hope it is clear now. I have attached a sample mark sheet as well so that you know how marks are…

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