STW 124MS-Assignment I (CW 1) PDF

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STW124MS – Logic and Sets Assignment Title: Coursework- Assignment I Intake: SEPTMEBER/NOVEMBER 2019 Importantnotes •

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Assignment Brief: 2019/20 Module Title: Logic and Sets

Ind

Cohort

Module Code

Sept 2019

STW124MS

Coursework Title Assignment I

Handout Date: 09/12/2019 Due Date

Lecturer: Shanta Rayamajhi Basnet

27/12/2019 (Before 5 PM) Estimated Time(hrs): 5 Word Limit:

Coursework Type: Written Assignment

% of Module Mark 40%

Submission arrangement To Softwarica Assigned Department File types and method of recording: Submission should be with one file Mark and Feedback date: 06/03/2020 Mark and Feedback method: Mark and Feedback file will be released on Moodle

Module Learning Outcomes Assessed: 1. Understanding of Propositional Logic 2. Understanding of Predicate Logic 3. Set , Relation and Function . Its Application and Database

Assignment – I Section: A

[40 Marks]

There are (EIGHT) 8 questions in this section, attempt ALL questions. Each question carries (FIVE) 5 marks.

Q.N.1 a. i. Find the truth value of ¬(X ∧ Y) ⇒ Z if X and Z are false, and Y is true. ii. What is the truth value if the brackets are removed?

b. Let p and q be the proposition p: Swimming is allowed. q: Sharks have been spotted near the shores Express each of the following compound propositions as an English sentence. q : p q: p  q : iii. p → q : iv.

i. ii.

c. Write the following statement in its equivalent converse, inverse and contrapositive form “if it rains, then I buy a new umbrella”.

(CLO1)

Q.N.2 Using laws of logic, prove that the following propositions are tautologies. a. [(p ʌ ¬q) ᴠ ¬p] ᴠ q b. [p ᴠ (¬ p ʌ q)] ᴠ (¬ p ʌ ¬ q)

(CLO1)

Q.N.3 A directory contains a set of large files, L = {f 1, f 2, f 3}, and set of data files, D = {d1, f 2, a1000}, and a set of text files T = {t1, t2, f 1, f 2}. a. Give a set expression for the set of large data files, and write the set out. b. Repeat for the set of text files which are not large. c. Repeat for the set of large data files which are plain text. d. Construct for the set of text files which are neither large nor data files

(CLO2)

Q.N.4 Given that universal set U = {x: x ≤ 15, x is a positive integer} A = {4, 6, 7, 8, 9, 13}, B = {the multiple of 4}, C = {6, 8, 9, 10, 12, and 15}. Then find a. A ∩ C b. B ∪ C  c. .A ∩B∩C d. A⊕ 𝐵 e. (A∪ B) ∩C (CLO2) Q.N.5 a. Express the following using the language of predicate calculus, where it is understood that the people being discussed is in the courtroom. If any sentence is ambiguous, give all possible symbolic versions. i. All judges are sober. ii. There is a dishonest lawyer. iii. All lawyers are honest. iv. All defendants are innocent. v. Some plaintiffs are lawyers.

b. Express the following in normal English: i. ∀x ∈ C : J(x) ∨ S(x) ii. ∀x ∈C : H(x) ∧ L(x) ⇒ S(x) iii. ∃x ∈C : P(x) ∧ D(x) c.Give the negation of each statement both in symbolic form and in natural English. i. All judges are sober. ii. There is a dishonest lawyer iii. All lawyers are honest.

(CLO2) Q.N.6 Construct a membership table for each of the following expressions: (a)  (X ∪ (Y ∪ Z)) ) (b) (X ∩ Y) ∪ (X ∩ 𝑌

(CLO2)

Q.N.7 a. What is the negation of the statements ∀𝑥 (𝑥 2 > 𝑥) 𝑎𝑛𝑑 ∋ 𝑥(𝑥 2 = 2)? b. What is truth value of ∀𝑥𝑃(𝑥),where P(x) is the statement “𝑥 2 < 10” and the domain consists of the positive integers not exceeding 4? (CLO2)

Q.N.8 Four Lectures Albert, Bertha, Harry are involved in teaching in degree program in Mathematics. Albert teaches Algebra, calculus and logic. Bertha teaches Algebra, calculus and Geometry. Harry teaches Algebra, Geometry and Differential Equations. Find the combinations of projection and inverse projection maps that calculate a. b. c. d. e.

The courses that Albert teaches. The lectures that teaches differential Equations. Those lecturers with course in common with Albert. Those courses taught by anybody who teaches Differential Equations. All those lectures teaching both algebra and Logic.

Section: B

(CLO2)

[60 Marks]

There are (SIX) 6 questions in this section, attempt ALL questions. Each question carries (TEN) 10 marks.

Q.N.1 Construct a truth table to establish the following compound propositions tautology, contradiction or contingency i. (p ∧ q) ∨ [~p∨ (p∧~q)] ii. [𝑝 → (𝑞∨r)]↔ [(𝑝∧~q) ∧~r]

(CLO1)

Q.N.2 a. let X= {e, g, h, j} and Y={5,6,7,8,9}. Define function f(e) = 9, f(g)= 6, f(h)=8, f(i)= 7. write the domain ,codomain and range of f. b. Given that f(x)=3x+2 and g(x) =𝑥 2 + 2𝑥 + 4 .Find i. fog(x)

ii. 𝑓 −1 (𝑥)

(CLO2)

Q.N.3 Let A= {1, 2, 3} and B = {a, b, c} b and C = {x, y, z} then consider R = {(1, b), (2, a), (2, c)} and S = {(a, y),(b, x),(c, y),(c, z)}.Then, a. b. c. d.

Find the composition of SoR Find the inverse of SoR Show the relation R on arrow diagram, matrix form and directed graph. Find 𝑀𝑅 , 𝑀𝑆 𝑎𝑛𝑑 𝑀𝑆0𝑅 and compare 𝑀𝑅 . 𝑀𝑆 𝑎𝑛𝑑 𝑀𝑆0𝑅 .

(CLO2)

Q.N. 4 a. Following a phone in survey on the use Web browsers and chat rooms of 135 people, the surveyors conclude that 69 of them use browsers, 84 of them use chat rooms to some extent, while 60 use browsers but not chat rooms. Do you believe these results? b. A company who wanted to develop its internet presence had 27 of its employees each investigate at least one of AOL, Demon, and BT. If 14 investigated AOL, 16 investigated BT, and 8 investigated Demon, while 1 investigated AOL and Demon, 5 investigated AOL and BT, and 5 investigated BT and Demon, how many investigated all three? (CLO2)

Q.N.5 a. State which rule of inference is used in the argument: If it rains today, then we will not have a barbecue today. If we don’t have a barbecue today, then we will have a barbecue tomorrow .Therefore, if it rains today, then we will have a barbecue tomorrow. b.Show that the hypothesis is valid: It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we don’t go swimming,then will take a canoe trip, if we take a canoe trip, then we will be home by sunset .Therefore we will be home by sunset. (CLO1)

Q.N.6 a. Sketch each of the following relations on {1, 2, 3, 4} × {a, b, c, d}: i. {(1, a),(2, a),(3, a),(4, a)} ii. {(1, a),(1, b),(2, c),(3, a),(4, a)} Explain which are functions, and classify the functions in terms of injectivity, surjectivity and bijectivity.

b.Let Q be the set of rational numbers. Prove that the function f:𝑄 → 𝑄suchthat f(x)= 3x+5 for all x∈ 𝑄 is one to one and onto function.Also find 𝑓 −1 (𝑥). (CLO2)

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