FOUNDATION OF COMPUTER SCIENCE ETCS-203

Transcription

1 ETCS-203 TUTORIAL FILE Computer Science and Engineering Maharaja Agrasen Institute of Technology, PSP Area, Sector 22, Rohini, Delhi

2 Fundamental of Computer Science (FCS) is the study of mathematical structures that are fundamentally discrete rather than continuous. FCS therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, Foundation of Computer Science has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term "Foundation of Computer Science." Indeed, FCS is also known as discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research. Discrete math is essential to college-level mathematics and beyond: Discrete math together with calculus and abstract algebra is one of the core components of mathematics at the undergraduate level. Students who learn a significant quantity of discrete math before entering college will be at a significant advantage when taking undergraduate-level math courses. Discrete math is the mathematics of computing: The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory. This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects. Indeed, at most universities, an undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree. We strongly recommend that, before students proceed beyond geometry, they invest some time learning elementary discrete math, in particular counting & probability and number theory. 2

3 Tutorial 1 1. Let X be the set of all students at a university. Let A be the set of students who are first year students, B the set of students who are second-year students, C the set of students who are in a discrete mathematics course, D the set of students who are international relations majors, E the set of students who went to a concert on Monday night, and F the set of students who studied until 2 AM on Tuesday. Express in set theoretic notation the following sets of students: (a) All second-year students in the discrete mathematics course. Sample Solution. {X E X : x E B and x c C}. (b) All first-year students who studied until 2 AM on Tuesday. (c) All students who are international relations majors and went to the concert on Monday night. (d) All students who studied until 2 AM on Tuesday, are second-year students, and are not international relations majors. (e) All first- and second-year students who did not go to the concert on Monday night but are international relations majors. (f) All students who are first-year international relations majors or who studied until 2 AM on Tuesday. (g) All students who are first- or second-year students who went to a concert on Monday night. (h) All first-year students who are international relations majors or went to a concert on Monday night. 2. Find at least two different ways to fill in the ellipses in the set descriptions given. For example, {2, 4..., 121 could be written either {2n : 1 < n < 6 and n G N} or {n+ I :n E {1,3,5,7, 111}. (a) {1, 3,..., 311 (b) 11, 2,..., 26) (c) {2, 5,..., Write three descriptions of the elements of the set {2, 5, 8, 11, 14). 4. How many elements does each of the following sets have? (a) A= 0 (b) B = {0} (c) C = {(0, 1), {1, 2)} (d) D - {0, 1, 2, {0, 1}, (1,21, {0, 1, 2}, A) (e) E = (0, {{1, {3, 5), {4, 5,7}, 8))) 5. Which of the following pairs of sets are equal? For each pair that is unequal, find an element that is in one but is not in the other. (a) (0, 1, 2) and {0, 0, 1, 2, 2, 11 (b) (0, 1, 3, 11,2}) and (0, 1, 2, (2, 3)) (c) {{1, 3, 5), {2, 4, 6), f5, 5, 1, 3}) and {{3, 5, 1}, {6, 4, 4, 4, 2), {2, 4, 4, 2, 6)) 3

4 (d) {{5, 3, 5, 1,51, (2, 4, 6), 15, 1, 3, 3)) and {{1, 3, 5, 1), {6, 4, 2), (6, 6, 4, 4, 6)) (e) 0 and{xe N:x > landx 2 =x) 6. This problem concerns the following six sets: A-={0,2,4,61 B-=11,3,5) C={0,1,2,3,4,5,6,7} D---0 E=EN F={{0,2,4,6)) (a) What sets are subsets of A? (b) What sets are subsets of B? (c) What sets are subsets of C? (d) What sets are subsets of D? (e) What sets are subsets of E? (f) What sets are subsets of F? 7. Let A={n:nn N and n=2k+1 for some ken},b={n :n EN and n= 4k + 1 for somek e N), and C = (MeE N m = 2k - 1 andk E N and k> 1). Prove the following: (a) 35 E A (b) 35 e C (c) 35 B (d) A = C (e) B C A (f) B C C (g) B C A (h) B C C 4

5 Tutorial 2 1. Let U={0,1,2,3,4,5, 6,7,8,9), A={0,1,2,3}, B={0,2,4}, and C= {0, 3, 6, 9). (a) FindAUB, AnB, A, (A n B), and (B U C) - A. (b) Find P(A), P(B), 7P(A n B), P(A) n P(B). (c) Is P(A U B) = P(A) U P(B)? Prove your answer. (d) Why doesn't P(A) make sense? 2. Let A = {0, 3) and B = {x, y, z}. Find the following: (a) A x B (b) A x A x B (c) B x A (d) B x A x B 3. Let A = {1, 2, {{l, 2}}}. (a) How many elements does A have? How many elements does 'P(A) have? How many elements does 7P(P (A)) have? In parts (b)-(m) determine, whether each of the following is true, and if not, explain why not. (b) I E A (c) {1,2leA (d) {{1,21) E A (e) 0EA (f) Ie EP(A) (g) {1,2} e P(A) (h) {{1,2)} e P(A) (i) 0 E P(A) (j) 1 E P(P (A)) (k) {1, 2} E P(P (A)) (1) {{1, 211 E P(P(A)) (m) 0 e P'(P(A)) 4. For each of the following statements, find the corresponding inverse, converse, and contrapositive. (a) If the stars are shining, then it is the middle of the night. (b) If the Wizards won, then they scored at least 100 points. (c) If the exam is hard, then the highest grade is less than 90. 5

6 Tutorial 3 1. In a class of 35 students who are either biology majors or have blonde hair, there are 27 biology majors and 21 blondes. How many biology majors must be blonde? 2. A film class had 33 students who liked Hitchcock movies, 21 students who liked Spielberg movies, and 17 students who liked both kinds of films. How many students were in the class if every student is represented in the survey? 3. A tennis camp has 39 players. There are 25 left-handed players and 22 players who have a two-handed back stroke. How many left-handed players have a two-handed back stroke if every player is represented in these two counts? 4. Prove by induction: There is a natural number k such that n! > n 3 for all n > k. 5 Let X = {a, b, c, d, e}. Let R, be the relation on X with elements {(a, b), (a, c), (d,e)}. Let R2 be the relation on X with elements {(a, b), (b, c), (c, d), (d, e), (e, a)}. For each of these relations, find the following: (a) The smallest relation on X that contains R and is reflexive (b) The smallest relation on X that contains R and is symmetric (c) The smallest relation on X that contains R and is transitive (d) The smallest relation on X that contains R and is reflexive and transitive 6 Let A = {1, 2, 3, 4}. Let the functions F, G, and H be given with domain and codomain A defined as F(1) = 3, F(2) = 2, F(3) = 2, and F(4) = 4 G(1) = 1, G(2) = 3, G(3) = 4, and G(4) = 2 H(1) = 2, H(2) = 4, H(3) = 1, and H(4) = 3 Find the following: (a) F o G (b) H o F (c) Go H (d) FoGoH 7 Let X = {1,2, 3, 4} and Y = {5, 6, 7, 8,9}. Let F = {(1, 5), (2,7), (4,9), (3, 8)}. Show that F is a function from X to Y. Find F - 1, and list its elements. Is F - 1 a function? Why, or why not? 6

7 Tutorial 4 Q1. Represent as propositional expressions: Tom is a math major but not computer science major P: Tom is a math major Q: Tom is a computer science major Use De Morgan's Laws to write the negation of the expression, and translate the negation in English Q2. Translate the sentences into propositional expressions: "Neither the fox nor the lynx can catch the hare if the hare is alert and quick." Q3. Write the contrapositive, converse and inverse of the expressions: P Q, ~P Q, Q ~P Q4. Determine whether the following arguments are valid or invalid: P Q,~P - ~Q P ~q, r q - ~p Q5. Find the truth table of the compound proposition (p q) (p r). Q6.Give the converse, the contrapositive, and the inverse of th e statement If it rains today, then I will drive to work. Q7. Show that p (q r) and q (p r) are logically equivalent using the laws of logical equivalences. Be sure to cite each law whenever used. 7

8 Q8. Use the table of logical equivalences to simplify the compound proposition [(p q) p] qbe sure to justify your answers Q9.Construct the truth table for ~(p q) (~r). Q10. Show that the statement (p q) [(~p) (~q)] is a contradiction. Q11. Show that the following are tautologies: (a) p (~p). (b) (p q) [(~p) (~q)] Q12. Obtain a disjunctive normal form of 1. (P Q) (~P ~Q) 2. P (Q R) v (P Q) Q13. Obtain a conjunctive normal form of 1. P v (Q R) 2. (~P R) (P Q) Q14. Obtain PCNF form of- 1. ~(P v Q) 2. ~(P Q) Q15. Obtain PDNF Form of Q (P v ~Q) 8

9 Tutorial 5 1. Using indirect method of proof, prove that ~p ^ ~q = ~ (p ^ q). 2. Show that the hypothesis x works hard, If x works hard, then he is dull boy and If x ix a dull boy, then he will not get a job imply conclusion x will not get a job. 3. Prove that the premises p q, q r, s r and p ^ s are inconsistent. 4. Prove that the premises a (b c), d (b ^ ~c) and (a ^ d) are inconsistent. 5. Using indirect method of Proof derive p ~s from the premises p (q v r), q ~p, s ~r, p. 6. Show that (p q) ^ (r s), (q t) ^ (s u), ~( t ^ u) and (p r) is equivalent to ~p. 7. Show that the following set of premises are inconsistent If Rama gets his degree, he will go for a job. If he goes for a job, he will get married soon. If he goes for higher study, he will not get married. Rama gets his degree and goes for higher study. 8. Show by indirect method of proof that for all x ( p(x) V q(x) ) is equivalent to ( for all x p (x) ) V ( there exist q (x)). 9. Prove the implication For all x ( p (x) q(x)), for all x ( r(x) q(x)) is equivalent to for all x ( r(x) ~ p(x)). 9