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This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on Discrete Mathematics Set 4
Q1 | (p → r) ∨ (q → r) is logically equivalent to
- (p ∧ q) ∨ r
- (p ∨ q) → r
- (p ∧ q) → r
- (p → q) → r
Q2 | Let P (x) denote the statement “x >7.” Which of these have truth value true?
- p (0)
- p (4)
- p (6)
- p (9)
Q3 | The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.
- ∃x(c(x) ∧ f (x))
- ∀x(c(x) ∧ f (x))
- ∃x(c(x) → f (x))
- ∀x(c(x) → f (x))
Q4 | The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.
- ∀x (f (x) → p (x))
- ∀x (f (x) ∧ p (x))
- ∃x (f (x) ∧ p (x))
- ∃x (f (x) → p (x))
Q5 | ”Everyone wants to learn cosmology.” This argument may be true for which domains?
- all students in your cosmology class
- all the cosmology learning students in the world
- both of the mentioned
- none of the mentioned
Q6 | Let domain of m includes all students, P(m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.
- a student is there who spends more than 2 hours in playing polo
- there is a student who does not spend more than 2 hours in playing polo
- all students spends more than 2 hours in playing polo
- no student spends more than 2 hours in playing polo
Q7 | Translate ∀x∃y(x < y) in English, considering domain as a real number for both the variable.
- for all real number x there exists a real number y such that x is less than y
- for every real number y there exists a real number x such that x is less than y
- for some real number x there exists a real number y such that x is less than y
- for each and every real number x and y such that x is less than y
Q8 | “The product of two negative real numbers is not negative.” Is given by?
- ∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
- ∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
- ∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
- ∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
Q9 | Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”
- ∀x l(x, joy)
- ∀y l(joy,y)
- ∃y∀x l(x, y)
- ∃x ¬l(joy, x)
Q10 | Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.
- all students does not like south indian dishes.
- amit does not like south indian people.
- amit does not like south indian dishes.
- amit does not like some dishes.
Q11 | Use quantifiers and predicates with more than one variable to express, “There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
- ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
- ∃x∃yp (x, y), where p (x, y) is “x has taken y,” the domain for x consists of all discrete maths lectures, and the domain for y consists of all pupil in this class
- ∀x∀yp(x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
- ∃x∀yp(x, y), where p (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all discrete maths lectures
Q12 | Find a counterexample of ∀x∀y(xy > y), where the domain for all variables consists of all integers.
- x = -1, y = 17
- x = -2 y = 8
- both x = -1, y = 17 and x = -2 y = 8
- does not have any counter example
Q13 | Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”
- modus tollens
- modus ponens
- disjunctive syllogism
- simplification
Q14 | Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”
- modus tollens
- conjunction
- hypothetical syllogism
- simplification
Q15 | Which rule of inference is used, ”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”
- simplification
- conjunction
- addition
- disjunctive syllogism
Q16 | What rules of inference are used in this argument?“All students in this science class has taken a course in physics” and “Marry is a student in this class” imply the conclusion “Marry has taken a course in physics.”
- universal instantiation
- universal generalization
- existential instantiation
- existential generalization
Q17 | What rules of inference are used in this argument?“It is either colder than Himalaya today or the pollution is harmful. It is hotter than Himalaya today. Therefore, the pollution is harmful.”
- conjunction
- modus ponens
- disjunctive syllogism
- hypothetical syllogism
Q18 | The premises (p ∧ q) ∨ r and r → s imply which of the conclusion?
- p ∨ r
- p ∨ s
- p ∨ q
- q ∨ r
Q19 | What rules of inference are used in this argument?“Jay is an awesome student. Jay is also a good dancer. Therefore, Jay is an awesome student and a good dancer.”
- conjunction
- modus ponens
- disjunctive syllogism
- simplification
Q20 | “Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that
- parul is out for trip
- raju is playing chess
- parul is out for a trip and raju is playing chess
- parul is out for a trip or raju is playing chess
Q21 | Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove
- ∀np ((n) → q(n))
- ∃ np ((n) → q(n))
- ∀n~(p ((n)) → q(n))
- ∀np ((n) → ~(q(n)))
Q22 | Which of the following can only be used in disproving the statements?
- direct proof
- contrapositive proofs
- counter example
- mathematical induction
Q23 | When to proof P→Q true, we proof P false, that type of proof is known as
- direct proof
- contrapositive proofs
- vacuous proof
- mathematical induction
Q24 | In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?
- direct proof
- proof by contradiction
- vacuous proof
- mathematical induction
Q25 | A proof covering all the possible cases, such type of proofs are known as
- direct proof
- proof by contradiction
- vacuous proof
- exhaustive proof