MCQOPTIONS
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MCQOPTIONS
This section includes 91 Mcqs, each offering curated multiple-choice questions to sharpen your Discrete Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
A monoid is called a group if _______a) (a*a)=a=(a+c)b) (a*c)=(a+c)c) (a+c)=ad) (a*c)=(c* |
| A. | (a*a)=a=(a+c) |
| B. | (a*c)=(a+c)c) (a+c)=ad) (a* |
| C. | (a*c)=(a+c)c) (a+c)=a |
| D. | (a*c)=(c*a)=e |
| Answer» E. | |
| 2. |
Condition for monoid is __________a) (a+e)=ab) (a*e)=(a+e)c) a=(a*(a+e)d) (a*e)=(e* |
| A. | (a+e)=a |
| B. | (a*e)=(a+e) |
| C. | a=(a*(a+e) |
| D. | (a* |
| Answer» E. | |
| 3. |
Consider an integer 23 such that 23 >= 3p for a 2p-cycle in a permutation group, then p is ___________ |
| A. | odd prime |
| B. | even prime |
| C. | rational number |
| D. | negative prime |
| Answer» B. even prime | |
| 4. |
A semigroup S under binary operation * that has an identity is called __________ |
| A. | multiplicative identity |
| B. | monoid |
| C. | subgroup |
| D. | homomorphism |
| Answer» C. subgroup | |
| 5. |
a * H is a set of _____ coset. |
| A. | right |
| B. | left |
| C. | sub |
| D. | semi |
| Answer» C. sub | |
| 6. |
Suppose that H be an X-set and suppose that a∼b and |Xₐ|=|Xᵦ|, the which of the following is true? |
| A. | Xₐ is powerset of Xᵦ |
| B. | Xₐ is isomorphic to Xᵦ |
| C. | Xₐ is homomorphic to Xᵦ |
| D. | Xᵦ is the subset of Xₐ |
| Answer» C. Xₐ is homomorphic to Xᵦ | |
| 7. |
_______ characterizes the properties of distributive lattices. |
| A. | Congruence Extension Property |
| B. | Algebraic extension property |
| C. | Poset |
| D. | Semigroup |
| Answer» C. Poset | |
| 8. |
If the 4 sides of a square are to be colored by colors. How many different colourings with 50 colours are there if two arrangements that can be obtained from each other by rotation are identical? |
| A. | 773762 |
| B. | 363563 |
| C. | 4536822 |
| D. | 1563150 |
| Answer» E. | |
| 9. |
What is an irreducible module? |
| A. | A cyclic module in a ring with any non-zero element as its generator |
| B. | A cyclic module in a ring with any positive integer as its generator |
| C. | An acyclic module in a ring with rational elements as its generator |
| D. | A linearly independent module in a semigroup with a set of real numbers |
| Answer» B. A cyclic module in a ring with any positive integer as its generator | |
| 10. |
The order of a simple abelian group is __________ |
| A. | infinite |
| B. | real number |
| C. | finite |
| D. | prime |
| Answer» B. real number | |
| 11. |
Let (z, *) is a group with x*y=x+y-2 then inverse of x is ___________ |
| A. | -(x+4) |
| B. | (x²+6) |
| C. | (x+y)/5 |
| D. | (3y+4x²) |
| Answer» B. (x²+6) | |
| 12. |
A trivial subgroup consists of ___________ |
| A. | Identity element |
| B. | Coset |
| C. | Inverse element |
| D. | Ring |
| Answer» B. Coset | |
| 13. |
What is a circle group? |
| A. | a subgroup complex numbers having magnitude 1 of the group of nonzero complex elements |
| B. | a subgroup rational numbers having magnitude 2 of the group of real elements |
| C. | a subgroup irrational numbers having magnitude 2 of the group of nonzero complex elements |
| D. | a subgroup complex numbers having magnitude 1 of the group of whole numbers |
| Answer» B. a subgroup rational numbers having magnitude 2 of the group of real elements | |
| 14. |
How many indistinguishable necklaces can be made from beads of 4 colors with exactly 9 beads of each color where each necklace is of length 16? |
| A. | 76967234 |
| B. | 5652209 |
| C. | 14414400 |
| D. | 8686214 |
| Answer» D. 8686214 | |
| 15. |
Let K be a group with 8 elements. Let H be a subgroup of K and H |
| A. | 8 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 16. |
_____ is the multiplicative identity of natural numbers. |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 17. |
A function is defined by f(x)=2x and f(x + y) = f(x) + f(y) is called _____________ |
| A. | isomorphic |
| B. | homomorphic |
| C. | cyclic group |
| D. | heteromorphic |
| Answer» B. homomorphic | |
| 18. |
Condition of semigroup homomorphism should be ____________ |
| A. | f(x * x) = f(x * y) |
| B. | f(x) = f(y) |
| C. | f(x) * f(y) = f(y) |
| D. | f(x * y) = f(x) * f(y) |
| Answer» E. | |
| 19. |
A function defined by f(x)=2*x such that f(x+y)=2x+y under the group of real numbers, then ________ |
| A. | Isomorphism exists |
| B. | Homomorphism exists |
| C. | Heteromorphic exists |
| D. | Association exists |
| Answer» C. Heteromorphic exists | |
| 20. |
A normal subgroup is ____________ |
| A. | a subgroup under multiplication by the elements of the group |
| B. | an invariant under closure by the elements of that group |
| C. | a monoid with same number of elements of the original group |
| D. | an invariant equipped with conjugation by the elements of original group |
| Answer» E. | |
| 21. |
An isomorphism of Boolean algebra is defined as _______ |
| A. | order isomorphism |
| B. | unordered isomorphism |
| C. | order homomorphism |
| D. | hyper-morphism |
| Answer» B. unordered isomorphism | |
| 22. |
The number of generators of cyclic group of order 219 is __________ |
| A. | 144 |
| B. | 124 |
| C. | 56 |
| D. | 218 |
| Answer» B. 124 | |
| 23. |
Let X be a n-square matrix such that Y = X + 8I. Which of the following property will exist? |
| A. | idempotent |
| B. | Y transpose is nilpotent |
| C. | X nilpotent |
| D. | Y inverse |
| Answer» C. X nilpotent | |
| 24. |
If the sum of elements in each row of an n×n matrix Z is zero, then the matrix is ______________ |
| A. | inverse |
| B. | non-singular |
| C. | additive inverse |
| D. | singular |
| Answer» E. | |
| 25. |
Lagrange’s theorem specifies __________ |
| A. | the order of semigroup is finite |
| B. | the order of the subgroup divides the order of the finite group |
| C. | the order of an abelian group is infinite |
| D. | the order of the semigroup is added to the order of the group |
| Answer» C. the order of an abelian group is infinite | |
| 26. |
The set of even natural numbers, {6, 8, 10, 12,..,} is closed under addition operation. Which of the following properties will it satisfy? |
| A. | closure property |
| B. | associative property |
| C. | symmetric property |
| D. | identity property |
| Answer» B. associative property | |
| 27. |
A group G, ({0}, +) under addition operation satisfies which of the following properties? |
| A. | identity, multiplicity and inverse |
| B. | closure, associativity, inverse and identity |
| C. | multiplicity, associativity and closure |
| D. | inverse and closure |
| Answer» C. multiplicity, associativity and closure | |
| 28. |
Let G be a finite group with two sub groups M & N such that |M|=56 and |N|=123. Determine the value of |M⋂N|. |
| A. | 1 |
| B. | 56 |
| C. | 14 |
| D. | 78 |
| Answer» B. 56 | |
| 29. |
A set of representatives of all the cosets is called _________ |
| A. | transitive |
| B. | reversal |
| C. | equivalent |
| D. | transversal |
| Answer» E. | |
| 30. |
Minimum subgroup of a group is called _____________ |
| A. | a commutative subgroup |
| B. | a lattice |
| C. | a trivial group |
| D. | a monoid |
| Answer» D. a monoid | |
| 31. |
If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________ |
| A. | * is associative |
| B. | * is commutative |
| C. | * is closure |
| D. | * is abelian |
| Answer» B. * is commutative | |
| 32. |
A subgroup has the properties of ________ |
| A. | Closure, associative |
| B. | Commutative, associative, closure |
| C. | Inverse, identity, associative |
| D. | Closure, associative, Identity, Inverse |
| Answer» E. | |
| 33. |
Consider the set B* of all strings over the alphabet set B = {0, 1} with the concatenation operator for strings ________ |
| A. | does not form a group |
| B. | does not have the right identity element |
| C. | forms a non-commutative group |
| D. | forms a group if the empty string is removed from |
| Answer» B. does not have the right identity element | |
| 34. |
Suppose Kₘ={P∈Sₘ|, |P| is odd prime}. Determine the set for which m≥3 Kₘ a subgroup of Sₘ. |
| A. | {3, 5, 7, 11, 13, …} |
| B. | {-14, -8, -3, 0, 3, 8, 14} |
| C. | {2, 4, 6, 8, 10, 12} |
| D. | {12, 25, 56, 78, 134,…} |
| Answer» B. {-14, -8, -3, 0, 3, 8, 14} | |
| 35. |
All the rings of order p2 is ____________ |
| A. | associative |
| B. | cyclic |
| C. | inverse |
| D. | commutative |
| Answer» E. | |
| 36. |
If Y⁹⁸ (a raised to the power of 5) = 0 and Y is a 97-square matrix. Determine the value of Y⁹⁷. |
| A. | I+Y |
| B. | -Y+3 |
| C. | 0 |
| D. | Y² |
| Answer» D. Y² | |
| 37. |
a * H = H * a relation holds if __________ |
| A. | H is semigroup of an abelian group |
| B. | H is monoid of a group |
| C. | H is a cyclic group |
| D. | H is subgroup of an abelian group |
| Answer» E. | |
| 38. |
If F is a free semigroup on a set S, then the concatenation of two even words is ________ |
| A. | a semigroup of F |
| B. | a subgroup of F |
| C. | monoid of F |
| D. | cyclic group of F |
| Answer» C. monoid of F | |
| 39. |
If X is an idempotent nonsingular matrix, then X must be ___________ |
| A. | singular matrix |
| B. | identity matrix |
| C. | idempotent matrix |
| D. | nonsingular matrix |
| Answer» C. idempotent matrix | |
| 40. |
In a group there must be only __________ element. |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 5 |
| Answer» B. 2 | |
| 41. |
An element of a commutative ring R(1≠0) is nilpotent if __________ |
| A. | a+1=0 |
| B. | aⁿ = 0, for some positive integer n |
| C. | aⁿ = 1, for some integer n |
| D. | a² = 0 |
| Answer» C. aⁿ = 1, for some integer n | |
| 42. |
B₁: ({0, 1, 2….(n-1)}, xₘ) where xₘ stands for “multiplication-modulo-n” and B₂: ({0, 1, 2….n}, xₙ) where xₙ stands for “multiplication-modulo-m” are the two statements. Both B₁ and B₂ are considered to be __________ |
| A. | groups |
| B. | semigroups |
| C. | subgroups |
| D. | associative subgroup |
| Answer» C. subgroups | |
| 43. |
The group of matrices with determinant _________ is a subgroup of the group of invertible matrices under multiplication. |
| A. | 2 |
| B. | 3 |
| C. | 1 |
| D. | 4 |
| Answer» D. 4 | |
| 44. |
A function f:(M,∗)→(N,×) is a homomorphism if ______ |
| A. | f(a, b) = a*b |
| B. | f(a, b) = a/b |
| C. | f(a, b) = f(a)+f(b) |
| D. | f(a, b) = f(a)*f(a) |
| Answer» C. f(a, b) = f(a)+f(b) | |
| 45. |
The dihedral group having order 6 can have degree _____________ |
| A. | 3 |
| B. | 26 |
| C. | 326 |
| D. | 208 |
| Answer» B. 26 | |
| 46. |
If (M, *) is a cyclic group of order 73, then number of generator of G is equal to ______ |
| A. | 89 |
| B. | 23 |
| C. | 72 |
| D. | 17 |
| Answer» D. 17 | |
| 47. |
An infinite cyclic group does not have a ______ series. |
| A. | AP |
| B. | GP |
| C. | Composite |
| D. | Finite |
| Answer» D. Finite | |
| 48. |
Suppose, M is a lower triangular matrix with all diagonal entries zero. The resultant matrix of M+I will be ___________ |
| A. | idempotent |
| B. | singular |
| C. | nilpotent |
| D. | inverse |
| Answer» C. nilpotent | |
| 49. |
Let H be a finite group. The order of Sylow p-subgroup of H for every prime factor p with multiplicity 9 is? |
| A. | p+6 |
| B. | p⁹ |
| C. | pᵖ |
| D. | 3!*p² |
| Answer» C. pᵖ | |
| 50. |
Every cyclic group is a/an ______ |
| A. | infinite subgroup |
| B. | abelian group |
| C. | monoid |
| D. | commutative semigroup |
| Answer» C. monoid | |