Discrete Mathematics MCQ PDF

# Eng : Mohamed Eldawansy ## Sections Mcq [sets and logic] [COMPANY NAME] | [Company address] ## Sections Mcq ### 1. A ______ is an ordered collection of objects. - a) Relation - b) Function - c) Set - d) Proposition **Answer: c** **Explanation: By the definition of set.** ### 2. The set O of od...

# Eng : Mohamed Eldawansy ## Sections Mcq [sets and logic] [COMPANY NAME] | [Company address] ## Sections Mcq ### 1. A ______ is an ordered collection of objects. - a) Relation - b) Function - c) Set - d) Proposition **Answer: c** **Explanation: By the definition of set.** ### 2. The set O of odd positive integers less than 10 can be expressed by - a) {1, 2, 3} - b) {1, 3, 5, 7, 9} - c) {1, 2, 5, 9} - d) {1, 5, 7, 9, 11} **Answer: b** **Explanation: Odd numbers less than 10 is {1, 3, 5, 7, 9}.** ### 3. Power set of empty set has exactly ______ subset. - a) One - b) Two - c) Zero - d) Three **Answer: a** **Explanation: Power set of null set has exactly one subset which is empty set.** ### 4. What is the Cartesian product of A = {1, 2} and B = {a, b}? - a) {(1, a), (1, b), (2, a), (b, b)} - b) {(1, 1), (2, 2), (a, a), (b, b)} - c) {(1, a), (2, a), (1, b), (2, b)} - d) {(1, 1), (a, a), (2, a), (1, b)} **Answer: c** **Explanation: A subset R of the Cartesian product A x B is a relation from the set A to the set B.** ### 5. The Cartesian Product B x A is equal to the Cartesian product A x B. Is it True or False? - a) True - b) False **Answer: b** **Explanation: Let A = {1, 2} and B = {a, b}. The Cartesian product A x B = {(1, a), (1, b), (2, a), (2, b)} and the Cartesian product B x A = {(a, 1), (a, 2), (b, 1), (b, 2)}. This is not equal to A x B.** ### 6. What is the cardinality of the set of odd positive integers less than 10? - a) 10 - b) 5 - c) 3 - d) 20 **Answer: b** **Explanation: Set S of odd positive an odd integer less than 10 is {1, 3, 5, 7, 9}. Then, Cardinality of set S = |S| which is 5.** ### 7. Which of the following two sets are equal? - a) A = {1, 2} and B = {1} - b) A = {1, 2} and B = {1, 2, 3} - c) A = {1, 2, 3} and B = {2, 1, 3} - d) A = {1, 2, 4} and B = {1, 2, 3} **Answer: c** **Explanation: Two set are equal if and only if they have the same elements.** ### 8. The set of positive integers is ______. - a) Infinite - b) Finite - c) Subset - d) Empty **Answer: a** **Explanation: The set of positive integers is not finite.** ### 9. What is the Cardinality of the Power set of the set {0, 1, 2}? - a) 8 - b) 6 - c) 7 - d) 9 **Answer: a** **Explanation: Power set P ({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0, 1, 2}) = {null, {0}, {1}, {2}, {0, 1}, {0,2}, {1, 2}, {0, 1, 2}}. ** ### 10. The members of the set S = {x | x is the square of an integer and x < 100} is - a) {0, 2, 4, 5, 9, 58, 49, 56, 99, 12} - b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81} - c) {1, 4, 9, 16, 25, 36, 64, 81, 85, 99} - d) {0, 1, 4, 9, 16, 25, 36, 49, 64, 121} **Answer: b** **Explanation: The set S consists of the square of an integer less than 10.** ### 1. The union of the sets {1, 2, 5} and {1, 2, 6} is the set ______. - a) {1, 2, 6, 1} - b) {1, 2, 5, 6} - c) {1, 2, 1, 2} - d) {1, 5, 6, 3} **Answer: b** **Explanation: The union of the sets A and B, is the set that contains those elements that are either in A or in B.** ### 2. The intersection of the sets {1, 2, 5} and {1, 2, 6} is the set ______. - a) {1 , 2} - b) {5, 6} - c) {2, 5} - d) {1, 6} **Answer: a** **Explanation: The intersection of the sets A and B, is the set containing those elements that are in both A and B.** ### 3. Two sets are called disjoint if there ______ is the empty set. - a) Union - b) Difference - c) Intersection - d) Complement **Answer: c** **Explanation: Two sets are disjoint if the intersection of two sets is the empty set.** ### 4. Which of the following two sets are disjoint? - a) {1, 3, 5} and {1, 3, 6} - b) {1, 2, 3} and {1, 2, 3} - c) {1, 3, 5} and {2, 3, 4} - d) {1, 3, 5} and {2, 4, 6} **Answer: d** **Explanation: Two sets are disjoint if the intersection of two sets is the empty set.** ### 5. The difference of {1, 2, 3} and {1, 2, 5} is the set ______. - a) {1} - b) {5} - c) {3} - d) {2} **Answer: c** **Explanation: The difference of the sets A and B denoted by A-B, is the set containing those elements that are in A not in B.** ### 6. The complement of the set A is ______. - a) A - B - b) U- A - c) A - U - d) B - A **Answer: c** **Explanation: The bit string for the union is the bitwise OR of the bit strings.** ### 10. The set difference of the set A with null set is ______. - a) A - b) null - c) U - d) B **Answer: a** **Explanation: The set difference of the set A by null set denoted by A - {null} is A.** ### 1. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then, the number of elements in A U B is ______. - a) 4 - b) 5 - c) 6 - d) 7 **Answer: a** **Explanation: AUB is {1, 2, 3, 4}.** ### 2. Let the set A is {1, 2, 3} and B is { 2, 3, 4}. Then, the number of elements in An Bis ______. - a) 1 - b) 2 - c) 3 - d) 4 **Answer: b** **Explanation: A ∩ B is {2, 3}.** ### 3. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then, the set A – B is ______. - a) {1, -4} - b) {1, 2, 3} - c) {1} - d) {2, 3} **Answer: c** **Explanation: In A – B the common elements get cancelled.** ### 4. In which of the following sets, A- B is equal to B - A - a) A= {1, 2, 3}, B ={2, 3, 4} - b) A= {1, 2, 3}, B ={1, 2, 3, 4} - c) A={1, 2, 3}, B ={2, 3, 1} - d) A={1, 2, 3, 4, 5, 6}, B ={2, 3, 4, 5, 1} **Answer: c** **Explanation: A- B= B-A = Empty set.** ### 5. Let A be the set of all prime numbers, B be the set of all even prime numbers, C be the set of all odd prime numbers, then which of the following is true? - a) A = BUC - b) B is a singleton set. - c) A = C U {2} - d) All of the mentioned **Answer: d** **Explanation: 2 is the only even prime number.** ### 6. If A has 4 elements B has 8 elements then the minimum and maximum number of elements in A U B are respectively ______. - a) 4, 8 - b) 8, 12 - c) 4, 12 - d) None of the mentioned **Answer: b** **Explanation: Minimum would be when 4 elements are same as in 8, maximum would be when all are distinct.** ### 7. If A is {{D}, {Φ, {O}}, then the power set of A has how many element? - a) 2 - b) 4 - c) 6 - d) 8 **Answer: b** **Explanation: The set A has got 2 elements so n(P(A))=4.** ### 8. Two sets A and B contains a and b elements respectively. If the power set of A contains 16 more elements than that of B, the value of 'b' and 'a' are respectively ______. - a) 4, 5 - b) 6, 7 - c) 2, 3 - d) None of the mentioned **Answer: a** **Explanation:32-16=16, hence a=5, b=4.** ### 9. Let A be {1, 2, 3, 4}, U be the set of all natural numbers, then U-A'(complement of A) is given by set ______. - a) {1, 2, 3, 4, 5, 6, ....} - b) {5, 6, 7, 8, 9, ......} - c) {1, 2, 3, 4} - d) All of the mentioned **Answer: c** **Explanation: U – A' = A.** ### 10. Which sets are not empty? - a) {x: x is a even prime greater than 3} - b) {x: x is a multiple of 2 and is odd} - c) {x: x is an even number and x+3 is even} - d) { x: x is a prime number less than 5 and is odd} **Answer: d** **Explanation: Because the set is {3}.** ### 1. The shaded area of figure is best described by _______. - a) A ∩ B - b) AUB - c) A - d) B **Answer: a** ### Mcq for Logic ### 1.1 Statement, Logical Connectives, Compound Statements and Truth Table ### 1. Which of the following is a statement in logic? - a) What a wonderful day! - b) Shut up! - c) What are you doing? - d) Bombay is the capital of India. **Answer: d** ### 2. Which of the following is a statement? - a) Open the door. - b) Do your homework. - c) Switch on the fan. - d) Two plus two is four. **Answer: d** ### 3. Which of the following is an open statement? - a) x+5=11 - b) Good morning to all. - c) What is your problem? - d) Listen to me, Rahul! **Answer: c** ### 4. Which of the following is not a proposition in logic. - a) √3 is a prime - b) √2 is an irrational number - c) Mathematics is interesting - d) 5 is an even integer **Answer: c** ### 5. Which of the following is a statement in Logic? - a) Go away - b) How beautiful! - c) x>5 - d) 2=3 **Answer: d** ### 6. Using quantifiers ∀, 3, convert the following open statement into true statement. 'x + 5 = 8, x ∈ N' - a) ∀x ∈ N, x + 5 = 8 - b) For every x ∈ N, x + 5 > 8 - c) ∃x ∈ N, such that x + 5 = 8 - d) For every x ∈ N, x + 5 < 8 **Answer: c** ### 7. ~(pv q) is - a) ~pvq - b) p→~q - c) p^~q - d) ~p^q **Answer: d** ### 8. If p: The sun has set, q: The moon has risen, then symbolically the statement 'The sun has not set or the moon has not risen' is written as ______. - a) p^~q - b) ~qvp - c) ~p^q - d) ~pv-q **Answer: b** ### 9. If p: Sita gets promotion, q: Sita is transferred to Pune. The verbal form of ~p ↔ q is written as ______. - a) Sita gets promotion and Sita gets transferred to Pune. - b) Sita does not get promotion then Sita will be transferred to Pune. - c) Sita gets promotion if Sita is transferred to Pune. - d) Sita does not get promotion if and only if Sita is transferred to Pune. **Answer: d** ### 10. p = There are clouds in the sky and q = it is not raining. The symbolic form is ______. - a) p→q - b) p^q **Answer: a** ### 11. Write in verbal form: p: he is fat, w: he is hard working, then (~p) v (~w) is ______. - a) If he is fat or he is hard working. - b) He is not fat and he is not hard working. - c) He is not fat or he is not hard working. - d) He is fat or hard working. **Answer: c** ### 12. If p: Rohit is tall, q: Rohit is handsome, then the statement 'Rohit is tall or he is short and handsome' can be written symbolically as ______. - a) p√(p^q) - b) p^(~pvq) - c) p√(p^~q) - d) ~p^(~p^~q) **Answer: a** ### 13. p: Sunday is a holiday, q: Ram does not study on holiday. The symbolic form of the statement 'Sunday is a holiday and Ram studies on holiday' is ______. - a) p^~q - b) p^q - c) ~p^~q - d) pv-q **Answer: b** ### 14. The converse of the statement 'If I work hard then I get the grade' is ______. - a) If I get the grade then I work hard. - b) If I don't work hard then I don't get the grade. - c) If I don't get the grade then I don't work hard. - d) If I work hard then I don't get the grade. **Answer: a** ### 15. The converse of 'If x is zero then we cannot divide by x' is ______. - a) If we cannot divide by x then x is zero. - b) If we divide by x then x is non-zero. - c) If x is non-zero then we can divide by x. - d) If we cannot divide by x then x is non-zero. **Answer: a** ### 16. Write verbally ~p ∨ q where p: She is beautiful; q: She is clever ______. - a) She is beautiful but not clever - b) She is not beautiful or she is clever - c) She is not beautiful or she is not clever - d) She is beautiful and clever. **Answer: b** ### 17. If p: Ram is lazy, q: Ram fails in the examination, then the verbal form of ~p v~q is ______. - a) Ram is not lazy and he fails in the examination. - b) Ram is not lazy or he does not fail in the examination. - c) Ram is lazy or he does not fail in the examination. - d) Ram is not lazy and he does not fail in the examination. **Answer: b** ### 18. The inverse of logical statement p → q is ______. - a) ~p→~q - b) p↔q - c) q→p - d) q↔p **Answer: a** ### 19. Let p: Mathematics is interesting, q: Mathematics is difficult, then the symbol p → q means ______. - a) Mathematics is interesting implies that Mathematics is difficult. - b) Mathematics is interesting is implied by Mathematics is difficult. - c) Mathematics is interesting and Mathematics is difficult. - d) Mathematics is interesting or Mathematics is difficult. **Answer: a** ### 20. Which of the following is logically equivalent to~(p^q) - a) p^q - b) ~pv-q - c) ~(pvq) - d) ~p^~q **Answer: b** ### 21. (pq) is equivalent to - a) p^~q - b) pvq - c) pv-q - d) ~p^~q **Answer: a** ### 22. Contrapositive of p → q is - a) q→p - b) ~q→p - c) ~q→~p - d) q→~p **Answer: c** ### 23. When two statements are connected by logical connective 'and', then the compound statement is called ______. - a) conjunctive statement. - b) disjunctive statement. - c) negation statement. - d) conditional statement. **Answer: a** ### 24. When two statements are connected by the connective 'if' then the compound statement is called ______. - a) conjunctive statement. - b) disjunctive statement. - c) biconditional statement. - d) conditional statement. **Answer: d** ### 25. For the statements 'p' and 'q' 'p→ q' is read as if p then q. Here, the statement 'q' is called ______. - a) antecedent. - b) consequent. - c) logical connective. - d) prime component. **Answer: b** ### 26. The contrapositive of the statement: "If a child concentrates then he learns" is ______. - a) If a child does not concentrate he can not learn. - b) If a child does not learn then he does not concentrate. - c) If a child practises then he learns. - d) If a child concentrates, he can't forget. **Answer: a** ### 27. A compound statement p or q is false only when ______. - a) p is false. - b) qis false. - c) both p and q are false. - d) depends on p and q. **Answer: c** ### 28. A compound statement p and q is true only when ______. - a) pis true. - b) q is true. - c) both p and q are true. - d) none of p and q is true. **Answer: c** ### 29. A compound statement p q is false only when ______. - a) p is true and q is false. - b) p is false but q is true. - c) atleast one of p or q is false. - d) both p and q are false. **Answer: a** ### 30. The statement, if it is raining then I will go to college' is equivalent to ______. - a) If it is not raining then I will not go to college. - b) If I do not go to college, then it is not raining. - c) If I go to college then it is raining. - d) Going to college depends on my mood. **Answer: a** ### 31. The converse of the statement "If Sun is not shining, then sky is filled with clouds" is ______. - a) If sky is filled with clouds, then the Sun is not shining. - b) If Sun is shining, then sky is filled with clouds - c) If sky is clear, then Sun is shining - d) If Sun is not shining, then sky is not filled with clouds **Answer: a** ### 32. Which of the following is the converse of the statement 'If Billu secures good marks, then he will get a bicycle'? - a) If Billu will not get bicycle, then he will secure good marks. - b) If Billu will get a bicycle, then he will secure good marks. - c) If Billu will get a bicycle, then he will not secure good marks. - d) If Billu will not get a bicycle, then he will not secure good marks. **Answer: b** ### 33. The contrapositive of the statement 'If Chandigarh is capital of Punjab, then Chandigarh is in India', is ______. - a) If Chandigarh is not in India, then Chandigarh is not a capital of Punjab - b) If Chandigarh is in India, then Chandigarh is capital of Punjab - c) If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India - d) If Chandigarh is capital of Punjab, then Chandigarh is not in India **Answer: a** ### 34. The connective in the statement "2 +7 > 9 or 2+7<9" is ______. - a) and - b) or - c) > - d) < **Answer: b** ### 35. The connective in the statement "Earth revolves round the Sun and Moon is a satellite of earth", is ______. - a) or - b) Earth - c) Sun - d) and **Answer: d** ### 36. The converse of the statement "If x > y, then x+a> y + a", is ______. - a) If xy, then x + a < y + a - b) Ifx+ay + a, then x > y - c) If xy, then x + a > y + a - d) If x>y, then x + a < y + a **Answer: b** ### 37. The statement "If x² is not even then x is not even", is the converse of the statement ______. - a) If x is odd, then x is even - b) If x is not even, then x² is not even - c) If x is even, then x² is even - d) If x is odd, then x² is even **Answer: c** ### 38. Every conditional statement is equivalent to ______. - a) its contrapositive - b) its inverse - c) its converse - d) only itself **Answer: a** ### 39. If p: Pappu passes the exam, q: Papa will give him a bicycle. Then the statement 'Pappu passing the exam, implies that his papa will give him a bicycle' can be symbolically written as ______. - a) p→q - b) p ↔ q - c) p^q - d) pvq **Answer: a** ### 40. The symbolic form of the statement 'Since it is raining the atmosphere is very cold' is ______. - a) p→q - b) p ↔ q - c) p^q - d) pvq **Answer: a** ### 41. Assuming the first part of each statement as p, second as q and the third as r, the statement 'Candidates are present, and voters are ready to vote but no ballot papers' in symbolic form is ______. - a) (pvq)^~I - b) (p^~q)^г - c) (~p^q)^~ - d) (p^q)^~I **Answer: d** ### 42. Assuming the first part of each statement as p, second as q and the third as r, the statement 'A monotonic increasing sequence which is bounded above is convergent' in symbolic form is ______. - a) (p^q) → I - b) (pvq) → I - c) (p^q)↔r - d) (pvq) →I **Answer: a** ### 43. Assuming the first part of each statement as p, second as q and the third as r, the statement 'If A, B, C are three distinct points, then either they are collinear or they form a triangle' in symbolic form is ______. - a) p↔ (qvI) - b) (p^q) → r - c) p→(qvI) - d) p→(q^) **Answer: c** ### 44. If d: Drunk, a: accident, translate the statement 'If the Driver is not drunk, then he cannot meet with an accident' into symbols. - a) ~a~d - b) ~d→~a - c) ~dла - d) a~d **Answer: b** ### 1.2 Statement Pattern and Logical Equivalence: Tautology, Contradiction, Contingency ### 45. Statement ~p → ~q = p ↔ q is ______. - a) a tautology - b) a contradiction - c) contingency - d) proposition **Answer: a** ### 46. Given that p is 'false' and q is 'true' then the statement which is 'false' is ______. - a) ~p-q - b) p→(q^p) - c) p→~q - d) q→~p **Answer: a**