Discrete Mathematics (useful for UGCNET,GATE ,APSET ,Engineering mathematic):Top 50 mcqs

Here are 50 MCQs on Discrete Mathematics (useful for APSET preparation):

1. Propositional Logic

1. The negation of (p ∧ q) is:
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q
C) p ∨ q
D) p ∧ ¬q

2. Which is a tautology?
A) p ∧ ¬p
B) p ∨ ¬p
C) p → ¬p
D) ¬p → p

3. The contrapositive of p → q is:
A) q → p
B) ¬p → ¬q
C) ¬q → ¬p
D) p ∧ q

4. If p is false and q is true, p → q is:
A) True
B) False
C) Undefined
D) Depends on p

5. (p → q) is equivalent to:
A) ¬p ∨ q
B) p ∨ q
C) ¬p ∧ q
D) p ∧ q

2. Predicate Logic

6. The negation of ∀x P(x) is:
A) ∀x ¬P(x)
B) ∃x ¬P(x)
C) ¬∃x P(x)
D) ∃x P(x)

7. Which is valid?
A) ∀x P(x) → P(a)
B) P(a) → ∀x P(x)
C) ∃x P(x) → ∀x P(x)
D) None

8. The domain of discourse refers to:
A) Variables
B) Constants
C) Set of possible values
D) Functions

9. Existential quantifier means:
A) For all
B) There exists
C) None
D) Exactly one

10. The statement “Some students are intelligent” is:
A) Universal
B) Existential
C) Conditional
D) Biconditional

3. Set Theory

11. If A ⊆ B and B ⊆ A, then:
A) A ∩ B = ∅
B) A = B
C) A ∪ B = ∅
D) A ≠ B

12. Power set of a set with n elements has:
A) n² elements
B) 2n elements
C) 2ⁿ elements
D) n! elements

13. De Morgan’s law:
A) (A ∪ B)’ = A’ ∩ B’
B) (A ∩ B)’ = A’ ∪ B’
C) Both A and B
D) None

14. If |A| = 3, number of subsets is:
A) 6
B) 8
C) 9
D) 12

15. Cartesian product A × B means:
A) Ordered pairs
B) Unordered pairs
C) Union
D) Intersection

4. Relations

16. A relation R on A is reflexive if:
A) (a,a) ∈ R
B) (a,b) ∈ R
C) (b,a) ∈ R
D) None

17. If R is symmetric and transitive and reflexive, then R is:
A) Partial order
B) Equivalence relation
C) Function
D) Injective

18. A partial order must be:
A) Reflexive, symmetric
B) Reflexive, antisymmetric, transitive
C) Symmetric, transitive
D) None

19. Hasse diagram represents:
A) Equivalence relation
B) Partial order
C) Function
D) Set

20. In an equivalence relation, sets formed are:
A) Subsets
B) Partitions
C) Ordered pairs
D) None

5. Functions

21. A function f: A → B is one-one if:
A) f(a1)=f(a2) ⇒ a1=a2
B) Every element of B has pre-image
C) Onto
D) Many-one

22. If every element of B has a pre-image, f is:
A) Injective
B) Surjective
C) Bijective
D) Partial

23. Composition of functions is:
A) f + g
B) f ∘ g
C) f × g
D) f − g

24. A bijective function has:
A) Inverse
B) No inverse
C) Only injective
D) Only surjective

25. Number of functions from set of size m to n is:
A) mⁿ
B) nᵐ
C) mn
D) m+n

6. Combinatorics

26. nPr equals:
A) n!/(n−r)!
B) n!/(r!(n−r)!)
C) r!
D) n!

27. nCr equals:
A) n!/(n−r)!
B) n!/(r!(n−r)!)
C) r!
D) n!

28. Number of ways to arrange n distinct objects:
A) n
B) n²
C) n!
D) 2ⁿ

29. Pigeonhole principle ensures:
A) At least one repetition
B) No repetition
C) Equal distribution
D) None

30. Binomial theorem expansion of (a+b)ⁿ has:
A) n terms
B) n+1 terms
C) n² terms
D) 2ⁿ terms

7. Recurrence Relations

31. Fibonacci sequence relation:
A) Fn = Fn−1 + Fn−2
B) Fn = 2Fn−1
C) Fn = Fn−1 − Fn−2
D) None

32. Linear recurrence has:
A) Constant coefficients
B) Variable coefficients
C) No coefficients
D) None

33. Characteristic equation is used for:
A) Non-linear recurrence
B) Linear homogeneous recurrence
C) Set theory
D) Graphs

34. Order of recurrence relation means:
A) Highest power
B) Number of previous terms
C) Degree
D) None

35. Base condition is also called:
A) Initial condition
B) Boundary
C) Relation
D) Proof

8. Graph Theory

36. A simple graph has:
A) No loops & no multiple edges
B) Loops allowed
C) Multiple edges allowed
D) Directed edges

37. Degree of a vertex in undirected graph counts:
A) Incoming edges
B) Outgoing edges
C) Total edges incident
D) None

38. In any graph, sum of degrees =
A) E
B) 2E
C) V
D) V²

39. A tree with n vertices has edges:
A) n
B) n−1
C) n+1
D) n²

40. A connected graph with no cycles is:
A) Tree
B) Complete
C) Bipartite
D) Null

9. Boolean Algebra

41. Idempotent law:
A) A + A = A
B) A + 0 = A
C) A·1 = A
D) A + A’ = 1

42. Complement law:
A) A + A’ = 1
B) A·A’ = 0
C) Both
D) None

43. Absorption law:
A) A + AB = A
B) A(A+B) = A
C) Both
D) None

44. Dual of A + 0 = A is:
A) A·1 = A
B) A+1=1
C) A·0=0
D) None

45. Boolean algebra is widely used in:
A) Databases
B) Digital circuits
C) Networking
D) AI

10. Number Theory

46. GCD of 12 and 18 is:
A) 2
B) 3
C) 6
D) 9

47. If a|b and b|c, then:
A) a|c
B) c|a
C) a+b
D) None

48. 17 is:
A) Composite
B) Prime
C) Even
D) Perfect

49. Euler’s totient function φ(p) for prime p equals:
A) p
B) p−1
C) 1
D) 0

50. Modular arithmetic deals with:
A) Division
B) Remainders
C) Factors
D) Roots

If you want, I can also provide:
✔️ Answer key
✔️ Detailed solutions
✔️ Model APSET practice paper (with answers)

Apset mcqs for DISCERETIVE mathematics 50 questions answers

Here are 50 APSET MCQs on Discrete Mathematics with Answers:

1. Propositional Logic

1. The negation of (p ∧ q) is:
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q
C) p ∨ q
D) p ∧ ¬q
Answer: B

2. Which is a tautology?
A) p ∧ ¬p
B) p ∨ ¬p
C) p → ¬p
D) ¬p → p
Answer: B

3. The contrapositive of p → q is:
A) q → p
B) ¬p → ¬q
C) ¬q → ¬p
D) p ∧ q
Answer: C

4. If p is false and q is true, p → q is:
A) True
B) False
C) Undefined
D) Depends on p
Answer: A

5. (p → q) is equivalent to:
A) ¬p ∨ q
B) p ∨ q
C) ¬p ∧ q
D) p ∧ q
Answer: A

2. Predicate Logic

6. The negation of ∀x P(x) is:
A) ∀x ¬P(x)
B) ∃x ¬P(x)
C) ¬∃x P(x)
D) ∃x P(x)
Answer: B

7. Which is valid?
A) ∀x P(x) → P(a)
B) P(a) → ∀x P(x)
C) ∃x P(x) → ∀x P(x)
D) None
Answer: A

8. The domain of discourse refers to:
A) Variables
B) Constants
C) Set of possible values
D) Functions
Answer: C

9. Existential quantifier means:
A) For all
B) There exists
C) None
D) Exactly one
Answer: B

10. “Some students are intelligent” is:
A) Universal
B) Existential
C) Conditional
D) Biconditional
Answer: B

3. Set Theory

11. If A ⊆ B and B ⊆ A, then:
A) A ∩ B = ∅
B) A = B
C) A ∪ B = ∅
D) A ≠ B
Answer: B

12. Power set of a set with n elements has:
A) n²
B) 2n
C) 2ⁿ
D) n!
Answer: C

13. De Morgan’s law states:
A) (A ∪ B)’ = A’ ∩ B’
B) (A ∩ B)’ = A’ ∪ B’
C) Both
D) None
Answer: C

14. If |A| = 3, number of subsets is:
A) 6
B) 8
C) 9
D) 12
Answer: B

15. Cartesian product A × B consists of:
A) Ordered pairs
B) Unordered pairs
C) Union
D) Intersection
Answer: A

4. Relations

16. A relation R on A is reflexive if:
A) (a,a) ∈ R for all a
B) (a,b) ∈ R
C) (b,a) ∈ R
D) None
Answer: A

17. A relation that is reflexive, symmetric and transitive is:
A) Partial order
B) Equivalence relation
C) Function
D) Injective
Answer: B

18. A partial order must be:
A) Reflexive, symmetric
B) Reflexive, antisymmetric, transitive
C) Symmetric, transitive
D) None
Answer: B

19. Hasse diagram represents:
A) Equivalence relation
B) Partial order
C) Function
D) Set
Answer: B

20. Equivalence relation partitions a set into:
A) Subsets
B) Partitions
C) Ordered pairs
D) None
Answer: B

5. Functions

21. A function is one-one if:
A) f(a1)=f(a2) ⇒ a1=a2
B) Every element of B has pre-image
C) Onto
D) None
Answer: A

22. A function is onto if:
A) Injective
B) Surjective
C) Bijective
D) None
Answer: B

23. Composition of functions is denoted by:
A) f + g
B) f ∘ g
C) f × g
D) f − g
Answer: B

24. A bijective function has:
A) Inverse
B) No inverse
C) Only injective
D) Only surjective
Answer: A

25. Number of functions from m elements to n elements:
A) mⁿ
B) nᵐ
C) mn
D) m+n
Answer: B

6. Combinatorics

26. nPr equals:
A) n!/(n−r)!
B) n!/(r!(n−r)!)
C) r!
D) n!
Answer: A

27. nCr equals:
A) n!/(n−r)!
B) n!/(r!(n−r)!)
C) r!
D) n!
Answer: B

28. Number of ways to arrange n distinct objects:
A) n
B) n²
C) n!
D) 2ⁿ
Answer: C

29. Pigeonhole principle states:
A) At least one box has ≥2 objects
B) No repetition
C) Equal distribution
D) None
Answer: A

30. (a+b)ⁿ has:
A) n terms
B) n+1 terms
C) n² terms
D) 2ⁿ terms
Answer: B

7. Recurrence Relations

31. Fibonacci recurrence is:
A) Fn = Fn−1 + Fn−2
B) Fn = 2Fn−1
C) Fn = Fn−1 − Fn−2
D) None
Answer: A

32. Order of recurrence relation means:
A) Highest power
B) Number of previous terms
C) Degree
D) None
Answer: B

33. Characteristic equation is used in:
A) Linear homogeneous recurrence
B) Set theory
C) Graph theory
D) None
Answer: A

34. A recurrence with constant coefficients is:
A) Linear
B) Non-linear
C) Variable
D) None
Answer: A

35. Initial conditions are also called:
A) Base conditions
B) Boundary
C) Solutions
D) None
Answer: A

8. Graph Theory

36. A simple graph has:
A) No loops and no multiple edges
B) Loops allowed
C) Multiple edges allowed
D) Directed edges
Answer: A

37. Degree of a vertex is:
A) Incoming edges
B) Outgoing edges
C) Total incident edges
D) None
Answer: C

38. Sum of degrees of all vertices equals:
A) E
B) 2E
C) V
D) V²
Answer: B

39. A tree with n vertices has:
A) n
B) n−1
C) n+1
D) n²
Answer: B

40. A graph with no cycles and connected is:
A) Tree
B) Complete
C) Bipartite
D) Null
Answer: A

9. Boolean Algebra

41. Idempotent law:
A) A + A = A
B) A + 0 = A
C) A·1 = A
D) A + A’ = 1
Answer: A

42. Complement law:
A) A + A’ = 1
B) A·A’ = 0
C) Both
D) None
Answer: C

43. Absorption law:
A) A + AB = A
B) A(A+B) = A
C) Both
D) None
Answer: C

44. Dual of A + 0 = A is:
A) A·1 = A
B) A+1=1
C) A·0=0
D) None
Answer: A

45. Boolean algebra is used in:
A) Databases
B) Digital circuits
C) Networking
D) AI
Answer: B

10. Number Theory

46. GCD(12,18) =
A) 2
B) 3
C) 6
D) 9
Answer: C

47. If a|b and b|c, then:
A) a|c
B) c|a
C) a+b
D) None
Answer: A

48. 17 is:
A) Composite
B) Prime
C) Even
D) Perfect
Answer: B

49. If p is prime, φ(p) =
A) p
B) p−1
C) 1
D) 0
Answer: B

50. Modular arithmetic deals with:
A) Division
B) Remainders
C) Factors
D) Roots
Answer: B

If you want, I can also prepare:
✔️ Hard level APSET MCQs (theory-based tricky questions)
✔️ Previous year pattern model paper (100 MCQs)
✔️ Topic-wise detailed explanation notes for APSET Discrete Mathematics

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