DISCRETE MATHEMATICS(PART-02)

 OBJECTIVE GENERAL KNOWLEDGE -  DISCRETE MATHEMATICS

1. A set of formulae A, B, C, D,......,R is said to be inconsistent if 
1. their disjunction is a tautology
2. their conjunction is a tautology
3. their conjunction is a contradiction
4. their disjunction is a contradiction
2. (x)(A(x) → B is equivalent to
1. (∃x) (A(x) → B)
2. ((∃x) A(x) → B)
3. ((x)(A(x)→B)
3. Which of the following propositions are equivalent over the universe of integers,.   (i) 0< n^2 ≤ 4. (ii)  0< n^3 ≤ 8. (iii)  0< n ≤ 2
1. (i) and (ii) are equivalent
2. (i) and (iii) are equivalent
3. all are equivalent
4. (ii) and (iii) are equivalent
4. 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.”
1. ∃x ¬L(Joy, x)
2. ∃y∀x L(x, y)
3. ∀x L(x, Joy)
4. ∀y L(Joy,y)
5. An Open statement is a declarative sentence
1. that is a tf- statement when each of its symbols is replaced by a specific object from a designated set
2. that has no symbols
3. that is a tf- statement
4. that is a closed statement
6. If P→(Q→S) 0s a valid from P→(Q→R) and Q→(R→S), then using CP rule, which is valid inference from the premises P, P→(Q→R) and Q→(R→S)
1. P→S
2. P→Q
3. Q→S
4. P→(Q→S)
7. 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.”
1. Existential specification
2. Existential generalization
3. Universal Specification
4. Universal generalization
8. 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.
1. No student spends more than 2 hours in playing polo
2. There is a student who does not spend more than 2 hours in playing polo
3. All students spends more than 2 hours in playing polo
4. A student is there who spends more than 2 hours in playing polo
9. “Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that
1. Parul is out for a trip or Raju is playing chess
2. Parul is out for a trip and Raju is playing chess
3. Parul is out for trip
4. Raju is playing chess
10. The truth value of ∀n(n + 1 > n) if the domain consists of all real numbers is true.
1. True
2. False
11. If monkeys can fly, then 1+1 = 3.
1. True
2. False
12. Using DeMorgan's law find the negation of the given statement : "Carlos will bicycle or run tomorrow"
1. Carlos will not bicycle tomorrow and carlos will not run tomorrow
2. Carlos will not bicycle tomorrow and carlos will run tomorrow
3. Carlos will bicycle tomorrow and carlos will not run tomorrow
13. “Match will be played only if it is not a humid day.” The negation of this statement is? (a) Match will be played but it is a humid day . (b) Match will be played or it is a humid day.
1. None of the statement
2. Both statement (a) and statement (b)
3. Statement (a) only
4. Statement (b) only
14. The relation "divides" on the set of all positive integers is
1. equivalence relation
2. Symmetric
3. Not reflexive
4. Not transitive
15. Let A= set of all real numbers, B = set of all positive real numbers; let f(a) = |a|, then inverse of f exist
1. True 
2. False
16. The recurrence relation for the Fibonacci sequence is
1. F(n)=F(n-1) - F(n-2)
2. No recurrence relation
3. F(n)=F(n-1) + 2
4. F(n)=F(n-1) + F(n-2)
17. The generating function for the sequence a[ n ] = 9Cn , n=0,1, 2, 3,.... is
1. (1+x)^9
2. (1+x)^-9
3. (1-x)^9
4. (1-x)^-9
18. Any connected bi-partite graph is uniquely
1. 2 - colourable
2. 3 - colourable
3. 5 - colourable
4. 4 - colourable
19. Every M-augmenting path is an M-alternating path
1. True
2. False
20. If a ≤≤ x, b ≤≤ x then
1. aΛb ≤ x aΛb ≤ x
2. aVb ≤ x aVb ≤ x
3. aΛb ≥ x aΛb ≥ x
4. aVb ≥ x aVb ≥ x
21. The chain(Z,≤)(Z,≤) contains
1. contains neither the least element nor the greatest element
2. greatest but not the least element
3. contains both the least and the greatest element
4. least but not the greatest element
22. The absorption law is
1. aΛa=aaΛa=a
2. aΛb=bΛaaΛb=bΛa
3. (aΛb)Λc=aΛ(bΛc)(aΛb)Λc=aΛ(bΛc)
4. aΛ(aVb)=aaΛ(aVb)=a
23. Let(L,≤)(L,≤) be a lattice . Assume aΛb=a aΛb=a
1. a = b
2. a≤b 
3. a≠b 
4. a≥b 
24. 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.”
1. ∃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.
2. ∃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.
3. ∃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.
4. ∀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.
25. Identify the which of the following is a tautology
1. ¬(x)A(x) = (∃x)(∃x) (¬A(x))
2. ¬(x)A(x) = (∃x)(∃x) (A(x))
3. ¬(x)A(x) = (x)(¬A(x))