Discrete Mathematics

Transcription

1 Discrete Mathematics Unit Logic Proofs Short Questions 1. Which of the following sentences are propositions (or statements)? What are the truth values of those that are propositions? (a) What a beautiful solar eclipse? (b) Every triangle is an equilateral triangle. (c) Please, st up. (d) Every quadrilateral is a square. (e) 4 + = 5. (f) 100. (g) Today is Thursday. (h) There is no pollution in Surat. (i) + 1 =. (j) The summer in India is hot sunny.. Define the following terms give the truth tables for each one by considering the propositions p q. (a) Conjunction Disjunction Connectives. (b) Conditional Bi-conditional propositions. (c) Exclusive Or. Define the following by giving an example. (a) Tautology Contradiction. (b) Dual of compound propositions. (c) Tautology implication. 4. When do you say that two compound propositions are logically equivalent? 5. What is the law of duality? 6. Define the following: (a) Disjunction Conjunctive normal forms of a proposition. (b) Truth value of a proposition. (c) Negation of propositions. (d) Converse of propositions. (e) Inverse of propositions. (f) Contrapositive of propositions. (g) Premise. (h) Conclusion. 7. How many rows appear in a truth table for each of these compound propositions? (a) (b) ( ) ( ) (c) (d) ( ) ( ) 8. Write down the duals of the following statements: (a) ( ) [ ] (b) ( ) (c) ( ) ( ) (d) ( ) ( ) 9. Let P(x) denote the statement 4 What are these truth values? (a) P(0) (b) P(4) (c) P(6) 1 Page

2 Discrete Mathematics Let P(x) be the statement =.If the domain consists of the integers, what are the truth values for the following? (a) ( ) (b) ( ) (c) P() Long Questions 1. State prove the following by using Truth Table. (a) Laws of Tautology Contradiction. (b) Commutative Laws. (c) Associative Laws. (d) De-Morgan s Laws. (e) Distributive Laws. (f) Law for Negation of a negation (or involution law) (g) Laws for absorption.. If t c denote tautology contradiction respectively p is a statement then prove. (a) = (b) = (c) (~ ) =. State the Primal Dual forms of the following Laws of Algebra of Propositions: (a) Idempotent Law (b) Identity Law (c) Dominant Law (d) Complement Law (e) Commutative Law (f) Associative Law (g) Distributive Law (h) Absorption Law (i) De-Morgan s Law 4. Prove the following equivalences by using the truth tables. Involving Conditionals: (a) (b) (c) (d) ( ) (e) ( ) (f) ( ) ( ) ( ) (g) ( ) ( ) ( ) (h) ( ) ( ) ( ) (i) ( ) ( ) ( ) Involving Biconditionals: (j) ( ) ( ) (k) (l) ( ) ( ) (m) ( ) 5. State Prove the following Tautological implications using Truth Tables: (a) (b) (c) (d) (e) ( ) (f) ) ( ) (g) ( ) (h) ( ) (i) ( ) Page

3 Discrete Mathematics (j) ( ) ( ) (k) ( ) ( ) ( ) 6. Find the converse, inverse contrapositive of each of these conditional statements. (a) If it snows today, I will ski tomorrow. (b) I come to class whenever there is going to be a quiz. (c) A positive integer is a prime only if it has no divisors other than 1 itself. (d) If it snows tonight, then I will stay at home. (e) I go to the beach whenever it is a sunny summer day. (f) When I stay up late, it is necessary that I sleep until noon. 7. Let P(x) be the statement x spends more than five hours every weekday in class, where the domain for x consists of all students. Express each of these quantifications in English. (a) ( ) (b) ( ) (c) ( ) (d) ( ) 8. What is the negation of each of the following propositions? (a) Today is Thursday (b) There is no pollution in New Jersey. (c) + 1 = (d) The summer in Kutch is hot sunny. 9. Let p, q, r be the propositions p: Grizzly bears have been seen in the area. q: Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, r logical connectives. (a) Berries are ripe along the trail, but grizzly bears have not been seen in the area. (b) Grizzly bears have not been seen in the area hiking on the trail is safe, but berries are ripe along the trail. (c) If berries are ripe along the trail, hiking is safe if only if grizzly bears have not been seen in the area. (d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area the berries along the trail are ripe. (e)for hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail for grizzly bears not to have been seen in the area. (f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area berries are ripe along the trail. 10. Find the bitwise OR, bitwise AND bitwise XOR of each of the following pairs of bit strings: (a) , (b) , (c) , Without using the truth table proves that ( ) ( ) is tautology. 1. Show that the following statements are equivalent: p:good food is not cheap. q:cheap food is not good. 1. Use the method of contraposition prove that, for any integer >, n is prime is odd. 14. Use the method of contradiction, prove that is not a rational number. True or False 1. If P(x) is true for all values of x in the universe of discourse is denoted by (x)p(x).. The truth value of (~ ) is tautology.. Let P(x) denotes the statement 4 then truth value of P(4) is F (false). 4. is equivalently written as ~. Page

4 Discrete Mathematics Unit- Basic Structures Short Questions 1. Define the following: (a) Set (b) Universal Set (c) Null Set (d) Singleton Set (e) Finite Set (f) Infinite Set (g) Cardinality or size of a set. (h) Subset of a set. (i) Superset (j) Proper Subset (k) Equal Sets. (l) Power Set (m) Ordered Pairs (n) Cartesian Product. (o) Union of two Sets. (p) Intersection of two Sets (q) Disjoint Sets. (r) Complement of a Set (s) Difference of a Set (t) Symmetric difference. Let A= {a, b,{c, d}, e}. How many elements does A contain?. Let S= {1,, } How many subsets does S contain? Which are they? S 4. Let S = {a, b}. How many elements does the power set contain? 5. Find the values of x y if the ordered pairs (x + y, 1) (7, x - y) are equal. 6. Suppose sets A, B C have, 4 elements respectively. How many elements are there in A B C? 7. Let D = {1, }, E = {, } F = {4, 5}. How many elements are there in (D E) F? Which are they? 8. From which sets does the Cartesian product {(, ); (, 4)} come from? 9. Explain the Roster notation set builder notation of sets with examples. 10. Define null set singleton set. 11. Define finite infinite sets. 1. What is cardinality of a set? 1. Define subset proper subset. 14. When are two sets said to be equal? 15. What is a power set? 16. State the relation between the cardinalities of a finite set its power set. 17. Define the Cartesian product of two sets give an example. 18. Define union intersection of two sets. Give their Venn diagram representation. 19. When are two sets said to be disjoint? 0. Define complement of a set. Give examples. 1. Define the symmetric difference of two sets.. What is the cardinality of a null set a singleton set? 4 Page

5 01 Discrete Mathematics If A = {a, b, c} B = {1, } then find A B B A. 4. Find the symmetric difference of {1,, 5} {1,, }? 5. How many different elements does A B have if A has m elements B has n elements? 6. Determine whether the sets are equal or not by giving appropriate reason. 7. What is the difference between proper improper subset? 8. What is the power set of empty set? 9. Find the symmetric difference of {,, 7 } {1, 9, }? 0. How many different elements does P(A) have if A has m elements? 1. For any sets A B, Is A B = B A. Justify your answer.. State the difference between Union Intersection of two sets.. If A= {1,, 5, 8} then find the Power sets of (i) A (ii). 4. How many different elements do B A have if A has elements B has 4 elements? 5. Let R be the set of real numbers let the function f : R R be defined by Then, find the values of f(5) f (6). 6. Let A = {Mary, Jane, Ann} B = {Pizza, Lasagne}. How many different functions are there from A into B? 7. The following diagram defines the function f, which maps the set {a, b, c, d, e} into itself. How many elements does the range of f contain? 8. Let the functions f, g, h be defined by, f(x) = x where x R, g (y) = y where < y < 9, h (z) = z where 1 < z < 10.Which of these functions are equal? 9. Let the functions f : R R g : R R be defined by f(x) = x; g(x) = Find the composition function, if exists, 40. Let the functions f : R R be defined by f (x) = x. Find f (7). 41. Define the following terms: (a) Domain of a function (b) Co-domain of a function (c) Range of a function (d) Function (e) Injective (One-one) function. (f) Surjective function (Onto) function. (g) Bijective function. (h) Transcendental functions (i) Algebraic functions (j) Polynomial functions (k) Rational functions (l) Irrational functions (m) Identity function (n) Floor function (o) Ceiling function (p) Integer-value function (q) Absolute value function (r) Remainder function (s) Composition of functions (t) Inverse of a function. 4. Give an example of a function, which is (a) Neither one-to-one nor onto. (b) One-to-one but not onto. (c) Onto but not one-to-one. 5 Page

6 01 Discrete Mathematics (d) One-to-one onto. 4. Show that the function given by is onto but not one-to-one. 44. Show that the function given by is onto but not one-to- one. 45. Check by giving reasons, whether the following are functions or not? (a) (b) (c) 46. Find the values of the following: (a) (b) (c). 47. State the difference between a geometric progression an arithmetic progression. 48. What are the values of these sums? 49. What are values of these sums, where S = {1,, 5, 7} 50. Compute each of these double sums. 51. What are the values of the following products? 5. Find the formula of the sequence with the first 10 terms is 5,11,17,,9,,41,47,5, Define the following terms: (a) Sequence (b) Geometric Progression (c) Arithmetic Progression (d) Permutation (e) Combination 54. State the Sum rule in Counting Principle. 55. State the Product rule in Counting Principle. 56. A new company with just two employees, A B, rents a floor of a building with 1 offices. How many ways are there to assign different offices to these two employees? 57. The chairs of an auditorium are to be labeled with a letter a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently? 58. What are the different circular arrangements of n objects? 59. There are microcomputers in a computer center. Each microcomputer has 4 ports. How many different ports to a microcomputer in the center are there? 60. How many different bit strings of length seven are there? 61. How many bit strings of length eight either start with a 1 bit or end with the two bits 00? 6. State the Pigeonhole Principle. 6. Among any group of 67 people, how many of them be at-least with the same birth-date? 64. In any group of 7 English words, how many must be at-least begin with the same letter? 65. List all the permutations of {a, b, c}. 66. How many different permutations are there of the set {a, b, c, d, e, f, g}? 67. How many permutations of {a, b, c, d, e, f, g} end with a? 68. Find the value of each of these quantities. (a) P(6, ); (b) P(6, 5); (c) P(8, 1); (d) P(8, 5); (e) P(8, 8); (f) P(10, 9). 69. Find the value of each of these quantities: (a) C(5, 1); (b) C(5, ); (c) C(8, 4); (d) C(8, 8); (e) C(8, 0); (f) C(1, 6) 70. Define a Countable Set. 71. When can we say that two sets have the same cardinality? 6 Page

7 Discrete Mathematics Long Questions 1. Let U = {1,,,..., 8, 9}, B = {1,, 5, 7} C = {,, 4, 5, 6}. How many elements does the sets ( ), ( ) contains? Also find the elements.. Given that U = {1,,,.,9,10}, A = {1,,,4,5}, B = {1,,4,8}, C = {1,,,5,7} D = {,4,6,8}, find each of the following: a) ( ) b) ( ) c) d) ( ) e) ( ) f) ( ) g) ( ) h) ( ) i) ( ) ( ) j) ( ) ( ) k) ( ) l) ( ). State the following algebraic laws of set theory with an example. Also, prove them analytically graphically. a) Identity law b) Domination law c) Idempotent law d) Inverse law e) Commutative law f) Associative law g) Distributive law h) De Morgan s law i) The principle of duality j) Absorption Law. 4. For any sets, A, B C, Prove that both analytically graphically that, (a) ( ) ( ) = (b) ( ) = ( ) ( ) (c) ( ) = ( ) ( ) (d) ( ( )) = ( ) 5. If A, B C are sets, prove analytically that ( ) = ( ) ( ) 6. If A, B, C D are sets, prove analytically that ( ) ( ) = ( ) ( ) 7. Verify De-Morgan s law = in set theory by using Venn diagram. If U=, A={1,,,4,5}, B={1,,4,8}, C={1,,,5,7} D={,4,6,8}, then find: (i) ( ( ) ) (ii) ( ) ( ). 8. If A = {1,, 5}, B = {6, 8, 10} then find the cardinality of the sets (a) (b) 9. Verify analytically ad graphically for sets A, B C ( ) =( ) ( ) 10. Construct the membership table to verify ( ) = for sets A B. 11. If U={x/x N, 1 x 10}, A={1,,,4,5}, B={1,,4,8}, C={1,,,5,7} D={,4,6,8}, then find: (i) ( ( ) ) (ii) ( ) ( ). 1. Verify analytically graphically for sets A, B C, ( ) =( ) ( ) 1. Construct the membership table to verify ( ) = ( ) for sets A, B C. 14. If U={x/x N, 1 x 10}, A={1,,,4,5}, B={1,,4,8}, C={1,,,5,7} D={,4,6,8}, then find: 7 Page

8 01 Discrete Mathematics (i) (ii). 15. Check whether the functions are bijective or not? (a), (b) 16. If A = {1,, } are functions from A to A given by find if exists, (i) ; (ii) ; (iii). 17. If is given by (a), find inverse of in each case. 18. If are defined by Show that are inverses of each other. 19. If S={1,,, 4} are defined by,, find if exists (a) (b) (c) (d) 0. Determine whether each of the following functions is an injective / or surjective function. (a), defined by (b) defined by (c) defined by 1. If A = {1,,, 4, 5}, B = {1,,, 8, 9} the functions are defined by find if exists.. If S={1,,, 4, 5} if the functions are given by,, (a) Verify whether (b) Explain why (c) Find. (d) Show that. If have inverses but are defined by doesnot., find check if they are equal. 4. Determine whether the function defined by is bijective or not. 5. Determine whether the function defined by is bijective or not. 6. Determine whether the function defined by is invertible or not. 7. Determine whether the function defined by is bijective or not. 8. Determine whether the function defined by is invertible or not. 9. If for, where R is the set of real numbers; prove that (a) (b) (c) (d). 0. Let A = {1,, } let f g be functions of A into A. Let f g be the set of points displayed in the first second diagram respectively. Find the value of f(g(1)). 1. If there are mathematics majors 5 computer science majors at a college, then find (a) How many ways are there to pick two representatives so that one is mathematics major the other is a computer science major? (b) How many ways are there to pick one representative who is either a mathematics major or a computer science major?. A multiple choice test contains 10 questions. There are five possible answers for each question. 8 Page

9 Discrete Mathematics (a) How many ways can a student answer the questions on the test if the student answers every question? (b) How many ways can a student answer the questions on the test if the student can leave answers blank?. By using Counting Principle, find the value of k after the following code has been executed. K = 0; For i1 = 1:5 For i = 1:4 For i = 1: For i4 = 1: K = k + 1; End; End; End; End; 4. By using Counting Principle, find the value of k after the following code has been executed. K = 0; For i1 = 1:5 K = k + 1; End; For i = 1:4 K = k + 1; End; For i = 1: K = k+1; End; For i4 = 1: K=k+1 End; 5. Prove that If a r are real numbers r 0, then 6. Assuming that repetitions are not permitted, (a) How many four digit numbers can be formed from the six digits 1,,, 5, 6 8? (b) How many of these numbers are less than 4000? (c) How many of the numbers in part (a) are even? (d) How many of the numbers in part (a) are odd? (e) How many of the numbers in part (a) are multiples of 5? (f) How many of the numbers in part (a) contain both the digits (a) In how many ways can 6 boys 4 girls sit in a row? (b) In how many ways can they sit in a row if the boys are to sit together the girls are to sit together? (c) In how many ways can they sit in a row if the girls are to sit together? (d) In how many ways can they sit in a row if just the girls are to sit together? 8. Suppose that either a member of the mathematics faculty or a student who is a mathematics major is chosen as a representative to a university committee. How many different choices are there for this representative if there are 7 members of the mathematics faculty 8 mathematics majors no one is both a faculty member a student? 9. A student can choose a computer project from one of three lists. The three lists contain, 15, 19 possible projects, respectively. No project is on more than one list. How many possible projects are there to choose from? 40. An office building contains 7 floors has 7 offices on each floor. How many offices are in the building? 41. In how many ways can we select three students from a group of five students to st in line for a picture? In how many ways can we arrange all five of these students in a line for a picture? 9 Page

10 Discrete Mathematics How many ways are there to select a first prize winner, a second prize winner, a third prize winner from 100 different people who have entered a contest? 4. Suppose that there are eight runners in a race. The winner receives a gold medal, the second place finisher receives a silver medal, the third place finisher receives a bronze medal. How many different ways are there to award these medals, if all possible outcomes of the race can occur there are no ties? 44. Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a Specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities? 45. How many poker hs of five cards can be dealt from a stard deck of 5 cards? Also, how many ways are there to select 47 cards from a stard deck of 5 cards? 46. A group of 0 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission (assuming that all crew members have the same job)? 47. Suppose that there are 9 faculty members in the mathematics department 11 in the Computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the mathematics department four from the computer science department? 48. How many positive integers n can be formed using the digits, 4, 4, 5, 5, 6, 7, if n has to exceed 50,00,000? 49. How many bit strings of length 10 contain (a) exactly four 1 s, (b) at-most four 1 s (c) at least four 1 s (d) an equal number of 0 s 1 s? 50. How many permutations of letters A B C D E F G contain (a) the string BCD, (b) the CFGA string, (c) the strings BA GF, (d) the strings ABC DE, (e) the strings ABC CDE, (f) the strings CBA BED? 51. If 6 people A, B, C, D, E, F are seated about a round table, how many different circular arrangements are possible, if arrangements are considered the same when one can be obtained from the other by rotation? If A, B, C are females the others are males, in how many arrangements do the genders alternate? 5. From a club consisting of a men 7 women, in how many ways can we select a committee of: (a) men 4 women? (b) 4 persons which has at-least one woman? (c) 4 persons that has at-most one man? (d) 4 persons that has persons of both genders? (e) 4 persons that so that two specific members are not included? 5. In how many ways can 0 students out of 0 be selected for an extra-curricular activity, if (a) Rama refuses to be selected? (b) Raja insists on being selected? (c) Gopal Govind insist on being selected? (d) Either Gopal or Govind or both get selected? (e) Just one of Gopal Govind gets selected? (f) Rama Raja refuse to be selected together? balls are to be placed in boxes. Each can hold all the 5 balls. In how many different ways can we place the balls so that no box is left empty, if (a) Balls boxes are different? (b) Balls are identical boxes are different? (c) Balls are different boxes are identical? (d) Balls as well as boxes are identical? 55. Show that the set of odd positive integers is a countable set. 56. Show that the set of all integers is countable. 57. Show that the set of positive rational numbers is countable. 58. Show that the set of real numbers is an uncountable set. 10 P a g e

11 Discrete Mathematics Multiple Choice Questions 1. Let A = {, {4, 5}, 4}. Which statement is correct? (a) 5 is an element of A. (b) {5} is an element of A. (c) {4, 5} is an element of A. (d) {5} is a subset of A..Which of these sets is finite? (a) {x x is even} (b) {x x < 5} (c) {1,,,...} (d) {1,,,...,999,1000}. Which of these sets is not a null set? (a) A = {x 6x = 4 x = 1} (b) B = {x x + 10 = 10} (c) C = {x x is a man older than 00 years} (d) D = {x x < x} 4. Let D E. Suppose Which of the following statements must be true? (a) c D (b) b D (c) a E (d) a D S 5. Which set S does the power set = {, {1}, {}, {}, {1, }, {1, }, {, }, {1,, }} come from? (a) {{1},{},{}} (b) {1,, } (c) {{1, }, {, }, {1, }} (d) {{1,, }} 6. Let A = {x, y, z}, B = {v, w, x}. Which of the following statements is correct? (a) A B = {v, w, x, y, z} (b) A B = {v, w, y, z} (c) A B = {v, w, x, y} (d) A B = {x, w, x, y, z} 7. Let A = {1,,,..., 8, 9} B = {, 5, 7, 9}. Which of the following statements is correct? (a) A B = {, 4, 6} (b) A B = {1,,, 4, 5, 6, 7, 8, 9} (c) A B = {1,, 4, 6, 8} (d) A B = {, 5, 7, 9} 8. Let C = {1,,, 4} D = {1,, 5, 7, 9}. How many elements does the set C D contain? How many elements does the set C D contain? 9. Let A = {,, 4}, B = {} C = {x x is even}. Which statement is correct? (a) C A = B (b) C B = A (c) A C (d) C / A = B 10. Let A B, B C D A = C. Which statement is always false? (a) B D (b) A C (c) A = B (d) B D = B A 11. What is shaded in the Venn diagram below? (a) A B (b) A (c) A (d) B 1. What is shaded in the Venn diagram below? B (a) A B (b) A' (c) A B (d) B - A 1. Let U = {1,,,..., 8, 9} A = {1,, 5, 7}. Find A'. (a) A' = {, 4, 6, 8} b) A' = {, 4, 6, 8, 9} (c) A' = {, 4, 6} (d) A' = {9} 11 P a g e

12 01 Discrete Mathematics Find the ordered pairs corresponding to the points A B, which appear in the coordinate diagram {1,, } {1,, } below. (a) A = (, 1), B = (1, ) (b) A = (1, ), B = (, 1) (c) A = (, 1), B = (1, ) (d) A = (, ), B = (, ) 15. Let A = {1,, }. Which of the following diagrams of A A is a function from A into A? (d) None of them. 16. Which of the diagrams defines a function of A = {a, b, c, d} into B = {1,, } (a) (b) (c) (d) 17. Let f 1, f, f, f 4, f 5 be functions of R into R let f 1(x) = x + x - 4. Which of these functions are equal to f 1? (a) f (x) = x (b) f (y) = y 4 (c) f 4(z) = z + z - 4 (d) f 5(x) = x + x 18. Which of the following functions is one-one? (a) To each show assign its first performance. (b) To each student assign his mentor. (c) To each pair of shoes assign its price. (d) To each school assign the number of computers it has. 19. Let A = {1,,, 4}. Let f, g h be functions of A into R. Which one of them is one-one? (a) f(1) =, f() = 4, f() = 5, f(4) = (b) g(1) =, g() = 4, g() = 5, g(4) = (c) h(1) =, h() =4, h() =, h(4) = (d) None of above 0. Let A = [, 1]. Which of these functions are bijective on A? 4 (a) f(x) = x (b) g (x) = x (c) h(x) = x (d) None of above 1. Let S = {a, b, c, d}. Which of the following sets of ordered pairs is a function of S into S? (a) {(a, b), (c, a), (b, d), (d, c), (c, a)} (b) {(a, c), (b, c), (d, a), (c, b), (b, d)} (c) {(a, c), (b, d), (d, b)} (d) {(d, b), (c, a), (b, e), a, c)}. Let R be the set of real numbers let the function f : R R be defined by f(x) = x -. Which ordered pair belongs to the graph f * of the function f? (a) (5, 6) (b) (, ) (c) (5, ) (d) (4, 1) True or False 1. If A is not a sub-set of B, then at least one element of A does-not belong to B. 1 P a g e

13 Discrete Mathematics Every set is a sub-set of itself.. If A is a subset of B, then is called the superset of A. 4. If S = {a, b, c}, then P(S) = The ordered pairs (a, b) (c, d) are equal, if only if a = b c = d. 6. A B is always equal to B A. 7. {, 6, 8} = {8,, 6} 8. {0} {0} 9. {(1, ), (, )} = {(, 1), (, )} 10. {5, 7, 8} = {8, 7, 5} 11. Trigonometric, exponential Logarithmic functions are Transcendental functions. 1. Identity function is always a bijective function = (mod 7) =. 15. The inverse of a function f, if exists, is always unique. Fill In the Blanks 1. is a subset of every set.. If A B B C, then A C.. If A = {a, b}, B = {a, b, c} C = {b, c, a} then A is a subset of C B is a subset of C. 4. Two sets A B are equal if,. 5. For any set S, if S = n then P(S) =. 6. For proving the algebraic laws of set theory, if we use definitions of sets, that method is called method if we use Venn diagram, then that method is called method. 7. A (B C) =. 8. Let f(x) = x. Then, the value of (5) =. 9. For any x set A, the function f: A A, f(x) = x is called the function on A. 10. A function of the form 4 ( ) ( ), where f(x) g(x) 0 are polynomials is called Function. 11. If f(x) = x x + x -4 is a function in x of degree =. 1. (mod 5) =. 14. ( ) =. Unit- Relations Short Questions 1. Determine whether or not each of the following relations is a function with domain {1,,, 4}.If any relation is not a function, explain why? (a) R1 = {(1, 1), (, 1), (, 1), (4, 1), (, )} (b) R = {(1, ), (, ), (4, )} (c) R = {(1, 1), (, 1), (, 1), (4, 1)} (d) R4 = {(1, 4), (, ), (, ), (4, 1)}. Determine whether or not each of the following relations is a function If any relation is a function, find its range. (a) R1 = {(x, y)/x, y Z, y = x + 7}, which is a relation from Z to Z. 1 P a g e

14 Discrete Mathematics (b) R = {(x, y)/x, y R, y = x}, which is a relation from R to R. (c) R = {(x, y)/x, y R, y = x + 1}, which is a relation from R to R. (d) R4 = {(x, y)/x, y Q, x + y = 1}, which is a relation from Q to Q.. Define the following: (a) Relation. b) Binary relation. (c) Domain of a relation. (d) Range of a relation. (e) Universal relation. (f) Void relation. (g) Identity relation. (h) Inverse of a relation (i) Intersection of two relations. (j) Union of two relations. (k) Difference of two relations. (l) Complement of a relation. (m) Reflexive property of a relation (n) Symmetric property of a relation (o) Anti-symmetric property of a relation. (p) Transitive property of a relation. (q) Equivalence relation. (r) Partial Order relation. (s) Poset. (t) Irreflexive relation. (u) Nodes or vertices in a graph. (v) Arcs or edges in a graph. (w) Directed graph or Diagraph of a relation. (x) Hasse Diagram. (y) n-array relation. (z) Composition of relation 4. If A = {,, 5} B = {6, 8, 10} arb if only if a A divides b B, then find R, R, D(R), D(R ), R(R) R(R ). 5. State an example of a relation for the following which satisfies: (a) Reflexive Property. (b) Symmetric property. (c) Anti-symmetric property (d) Transitive property (e) Equivalence relation property (f) Partial Order relation property (g) Irreflexive relation property (h) Non symmetric property (not symmetric) (i) Non anti-symmetric property (not anti-symmetric) (j) Non Transitive property (not transitive) (k) an n-array relation. 6. State the difference between an Equivalence relation a partial order relation. 7. State the difference between a set a Poset. 8. State an example of a Partially ordered set. 9. List the ordered pairs in the relation R from A = {0, 1,,, 4} to B = {0, 1,, }, where (a, b) R if only if : (i) a = b; (ii) a + b = 4; (iii) a > b; (iv) a/b (a divides b); (v) gcd(a, b) = 1; (vi) lcm(a, b) = 10. The relation R on the set A = {1,,, 4, 5} is defined by the rule (a, b) R, if divides a-b. (i) List the elements of R, R complement of R. (ii) Find the domain range of R R. 14 P a g e

15 Discrete Mathematics If R be a relation on Z Z Z consisting of triples (a, b, c), where a, b, c are integers with a b c. Is (-,, 7) (,, ) belongs to relation R or not? 1. Let R be the relation on set {1,,, 4} containing the ordered pairs {(1, 1), (1, ), (, 1), (, ), (, ), (, ), (4, 1), (4, 4)}. Find the reflexive closure Symmetric closure of R. 1. If R is the relation of A = {1,,,4} such that (a,b) R, if only if a + b = even, Find the relational matrix MR. 14. If R be a relation defined as, R = {(a, b)/a b} on the set {1,,, 4, 5, 6}. Find R. 15. If R be a relation defined as, R = {(a, b)/ a divides b} on the set {1,,, 4, 5, 6}. Find R. 16. If R be a relation on Z Z Z consisting of triples (a, b, c), where a, b, c are integers with a > b > c. Is (-,, 7) (, 4, 5) belongs to relation R or not? 17. Two sets E = {, 4, 5, 6} F = {1,, 6}. Let R be a relation from E to F, defined by " greater than " relation. How many elements does the solution set of R contains? 18. If R is the relation from A = {1,,,4} to B = {,,4,5}, list the element in R defined by arb, if a b are both odd. Write also the domain range of R. 19. If R is a relation from A = {1,,} to B = {4,5} given by R = {(1,4), (,4), (1,5), (,5)}. Then, find R (the inverse of R) R (the complement of R). 0. If R = {(1,1), (,), (,)} S = {(1,1), (1,), (1,), (1,4)} find R S. 1. Give an example of a relation that is both symmetric antisymmetric.. Give an examples of a relation that is neither symmetric nor antisymmetric.. Give an example of a relation that is reflexive transitive but not symmetric. 4. Give an example of a relation that is symmetric transitive but not reflexive. Long Questions 1. State the types of relations with an appropriate example of each one.. Let A = {x, y, z}, B = {1,, }, C = {x, y} D = {, }. Let be the relation from A to B defined by R = {(x, 1), (x, ), (y, )} Let S be the relation from C to D defined by S = {(x, ), (x, )}. Then, find R-S, R S, R S, R.. State the properties of relations with an appropriate example of each one. 4. Check whether the greater than or equal to ( ) relation is a partially ordered set or not in set of integer Z. 5. Construct a diagraph representing the relation R R on the set A = {,, 4, 6}, defined barb if a divides b, for every a, b A. 6. State the three steps to reduce a diagraph into a Hasse diagram. 7. Construct a diagraph hence, Hasse diagram representing the relation R on the set A = {1,,, 4}, defined by arb if a b, for every a, b A. 8. If R = {(1, ), (, 4), (, )} S = {(1, ), (, 4), (4, )}. Find the following: (i) R S; (ii) R S; (iii) R-S; (iv) S-R; (v) R S. Also, verify that, dom(r S) = dom(r) dom(s) range (R S) range(s). 9. Determine whether the relation R on the set of all integers is reflexive, symmetric, anti-symmetric or transitive or not, where arb if only if (i) a b; (ii) ab 0; (iii) ab 1; (iv) a is a multiple of b. (v) a - b = 1 ; (vi) a = b ; (vii) a b. 10. Which of the following relations on set {0, 1,, } are equivalence relations? Explain with appropriate reason. (i) R1 = {(0, 0), (1, 1), (, ), (, ) } (ii) R = {(0, 0), (0, ), (, 0), (, ), (, ), (, ), (, )} (iii) R = {(0, 0), (1, 1), (1, ), (, 1), (, ), (, )} (iv) R4 = {(0, 0), (1, 1), (1, ), (, ), (, ), (, 1), (, ), (, )} (v) R5 = {(0, 0), (0, 1), (0, ), (1, 0), (1, 1), (1, ), (, 0), (, ), (, )} 11. Construct a diagraph hence, Hasse diagram representing the relation R on the set A = {0, 1,,, 4}, defined by arb if, a b for every a, b A. 15 P a g e

16 Discrete Mathematics Construct a diagraph hence, Hasse diagram representing the relation R on the set A = {1,,,4,5,6,7,8} defined by arb if, a b for every a, b A. 1. Prove that the relation of set inclusion is a partial ordering on any collection of sets. 14. Construct a diagraph hence, Hasse diagram representing the relation R on the set A = {1,, 6, 7, 8, 9}, defined by arb if, a b for every a, b A. Also, by diagraph, check whether, the given relation is a partial order relation or not? Determine whether the relation represented by the zero-one matrix is an equivalence relation or not? 16. Define a relation on a set. Find the transitive closure of the relation R = {(1, 1), (1, ), (, 1), (, ), (, ), (, 1), (, )} on {1,, }. 17. Let R be the relation on the set {0, 1,, } containing the ordered pairs (0, 1), (1, 1),(1, ), (, 0) (, ) (, 0). Find the reflexive closure symmetric closure of R. 18. Find the matrix representing the relations SoR, where the matrices representing R S are MR = MS = Determine whether the relation represented by the zero-one matrix is an equivalence relation or not? 0. Construct a diagraph hence, Hasse diagram representing the relation on the set A = {1,, 4, 5, 6, 7}, defined by greater than or equal to relation. Also, by diagraph, Check whether, the given relation is a partial order relation or not? 1. Define a void relation on a set. Find the transitive closure of the relation R = {(1, 1), (1, ), (, 1), (, ), (, ), (, 1), (, ) } on {1,,}.. Define an Equivalence relation. Check whether the relation R on the set of all integers defined by arb if only if a b is an Equivalence relation or not? Multiple Choice Questions: 1. If the relation on a set is called a void relation if (a) R = A B (b) R = Ø (c) R = {(a, a) / a A} (d) R = R (e) None of above. If A = {1,, }, then R = {(1, 1), (, ), (, )} is the relation on A. (a) Void (b) Inverse (c) Identity (d) Universal (e) None of above. If R S denote two relations, then R-S is defined as, (a) ( ) = ~ ~ (b) ( ) = ~ (c) ( ) = ~ (d) ( ) = ~ ~ (e) None of above 4. To reduce a diagraph into a Hasse diagram, it satisfies (a) Since the partial ordering is a reflexive relation, its diagraph has loops at all the vertices. We need not show loops since they must be present. (b) Since, the partial ordering is a transitive, we need not show those edges that must be present due to transitivity (c) If we assume that all edges are directed upward, we need not show the directions of the edges. (d) All of above. (e) None of above. 16 P a g e

17 Discrete Mathematics The relation R is given by the set of ordered pairs, R = {(, 4), (, 4), (1, ), (, 5), (, )}. Which of the following is the domain of R? (a) {,, 1, 5} (b) {1,, } (c) {1,,, 4, 5} (d) {, 4,, 5} 6. The relation R is given by the set of ordered pairs, R = {(, 4), (, 4), (1, ), (, 5), (, )}. Which of the following is the range of R? (a) {, 4, 5} (b) {1,, } (c) {1,,, 4, 5} (d) {4, 4,, 5} 7. Find the inverse of the relation R = {(, 4), (, 4), (1,), (, 5), (, )}. (a) {(, 4), (, 4), (1, ), (, 5), (, )} (b) {(4, ), (4, ), (, 1), (5, ), (, )} (c) {5, 4,,, 1} (d) {(4, 4), (, ), (1, 1), (5, 5), (, )} 8. Which one of the following open sentences defines a reflexive relation on the set of natural numbers? (a) "x is less than y" (b) "x = y" (c) "x - y = 5" (d) "x divides y" 9. Find the reflexive relation on the set A if A = {a, b, c}. (a) R1 = {(a, b), (b, c), (a, a), (c, c)} (b) R = {(a, a), (b, c), (c, c)} (c) R = {(a, a), (a, c), (c, a), (b, b), (c, c)} (d) R4 = {(a, a), (c, c), (a, c), (c, a)} 10. Which one of the following open sentences defines a symmetric relation in the set of natural numbers N? (a) "x is less than y" (b) "xy = 1" (c) "x - y = 5" (d) "x divides y" 11. Find non-symetric relation on the set B = {c, d, e}. (a) R1 = {(e, e)} (b) R = {(e, e),(c, d), (d, c)} (c) R = B x B (d)r4 = {(c, d), (d, d), (d, e)} 1. Find the anti-symmetric relation on the set B = {1,, }. (a) R1 = {(, )} (b) R = {(1, ), (1, 1), (1, ), (, 1)} (c) R = B B (d) R4 = {(1, ), (, ), (, ), (, )} 1. Let C = {1,,, 4} let R1, R, R, R4 be the relations in C. Which one of them is transitive? (a) R1 = {(, ), (, ), (, ), (1, 1)} (b) R = {(1, 1), (, ), (1, ), (, )} (c) R = {(, 4), (, ), (4, 4), (4, )} (d) R4 = {(1, ), (, ), (, ), (, )} 14. Find the transitive relation in the set D = {1,,, 4}. (a) R1 = {(1, ), (4, ), (, ), (, 4)} (b) R = {(1, 4), (4, ), (1, 1), (, )} (c) R = {(, ), (, 4), (4, 4), (4, )} (d) R4 = {(, 1), (, 4), (4, ), (, 4)} 17 P a g e

18 Discrete Mathematics Unit-4 Matrices Short Questions 1. Define the following terms: a) Matrix b) Diagonal matrix c) Equal matrices d) Transpose of a matrix e) Row matrix f) Column matrix g) Null matrix h) Square matrix i) Symmetric matrix k) Skew-symmetric matrix l) Unit matrix OR identity matrix m) Determinant of square matrix. If A is a matrix of order p q B is a matrix of order q r, then what is the order of the product of matrix AB?. What is the necessary condition for the addition of two matrices? 4. What is the necessary condition for the multiplication of two matrices? 5. Every identity matrix is a diagonal matrix True or False? Justify your answer. 6. State the difference between a matrix a determinant If A = B =, find B A If Q = [ 1] S =, find QS. 9. If B =, find BT Find ad-joint of the following matrices: a) b) Find inverse of the following matrices: a) b) c) Long Questions 1. Find the elements a1, a, a, a from the following matrices: 18 P a g e

19 Discrete Mathematics The following matrices are of special types. Indicate their types also give their orders.. If possible, find A + 5B; A 4C; B + C -D; A + C; B D. 4. If A = ; B = A; C = B + A 5I (where I is an Identity matrix of order );find the matrix such that D = A + B C. T 5. If A = ; find the matrix B such that A + B = O; where O is the null matrix. 6. Find A + B C mention the type of the matrix obtained. 7. Find AB BA for the given matrices, 8. If A = ;B= 9. If A = ;C= ; B= Prove that A (BC) = (AB) C. ;C= Prove that: i. AB = BA = O (Null matrix) ii. AC = A iii.ca = C 10. If A = ; B= ; Find AB. 19 P a g e

20 Discrete Mathematics If A = Prove that A = I ; (where I is an Identity matrix of order ). 1. Prove that A = satisfies the following equation: A 4A - 5 I = O (where O is a null matrix of order I is an Identity matrix of order ). 1. If A = T B = find AB BB. 14. If A = ;B=. Prove that (A + B) = A + B. 15. If A = ;B= (A + B) = A + B, find x y. 16. Find the transpose of matrices A B; 17. If A= ;B=, Verify that, T T T i. (A + B) = A + B T T T ii. (AB) = B A 18. Find ad-joint of the following matrices: a) ; b) ; c) ; d) 19. Find inverse of the following matrices: 0. Examine whether the following matrices are non-singular or not. 1. Find also the inverse of a non-singular matrix. T. If A =, find A + A + A.. If A =,B= 4. If A = AB =, Show that (AB) = B A., find matrix B. 5. Solve the system of equations by Matrix Inversion method. (a) x + y = 7 (b) 5x y = 4x y = 9-6x + y = 4 (c) 5x + 8y + z = (d) 4x -y - z = y + z = -x+4y - z = x + y z = -x - y +5z = 0 0 P a g e

21 Discrete Mathematics Multiple Choice Questions: 4 6, the order of matrix A is 8 5 a) b) c) 1 d) None of the above. If B = 7, which type of the give matrix B? 1 a) Unit matrix b) Row matrix c) Column matrix d) Square matrix 1 4. Mention the type of the matrix A = a) Symmetric matrix b) Skew-symmetric matrix c) Null matrix d) Identity matrix is matrix of the type. 0 1 a) Zero matrix b) Row matrix c) Column matrix d) Unit matrix If A = 0 6 9, the diagonal elements are 7 8 a) 4, 6, 8 b) 4, 0, c), 6, d) All of the above 1. If A = 6. If A = a) ,B= 5 b) then, A B gives c) If B = 4 6 then co-factor of 6 is a) - b) c) 6 d) If A =,B=, the product BA is 1 1 a) b) c) If A =, then find adj A a) b) c) If B =, then the transpose of B is a) b) c) T T 11. If A =, find (A ) a) b) c) 5 1. If A =, then A 5 a) b) c) d) d) None of the above. d) d) d) None of the above. d) 1 P a g e

22 Discrete Mathematics If A = 1 a) I 14. If B = a) B 15. If A = a) , then A = 0 b) I4 c) I 01 d) none of the above then, adj (adj (B) ) gives b) adj (B) 6, then A = 4 b) 1 c) B d) c) 0 d) None of the above P a g e