Module - 3

1. What is a cyclic group

A. A group in which the binary operation is commutative

B. A group that contains only one element

C. A group in which every element can be generated by a single element

D. A group that does not have an identity element

2. The intersection of two subgroups of a group is always a:

A. Subgroup

B. Supergroup

C. Coset

D. Non-group

3. The identity element in a monoid is:

A. The element that absorbs all other elements

B. The element that is not part of the set

C. The element that has an inverse

D. The element that does not affect other elements

4. A binary operation is said to be commutative if:

A. a * b = 1 for all values of a and b

B. a * b = a for all values of a and b

C. a * b = b for all values of a and b

D. a * b = b * a for all values of a and b

5. The binary operation "⊙" is associative if:

A. (a ⊙ b) ⊙ c = a ⊙ (b ⊙ c) for all values of a, b, and c

B. (a ⊙ b) ⊙ c = a ⊙ b for all values of a, b, and c

C. (a ⊙ b) ⊙ c = c for all values of a, b, and c

D. (a ⊙ b) ⊙ c = 1 for all values of a, b, and c

6. Every cyclic group is also a:

A. Generator

B. Abelian group

C. Non-abelian group

D. Trivial group

7. In a monoid, the binary operation must be:

A. Distributive

B. Commutative

C. Associative

D. Inverse

8. In a semigroup, the binary operation must be:

A. Associative

B. Commutative

C. Distributive

D. Inverse

9. A binary operation is said to be distributive if:

A. a * (b + c) = a * b + a * c for all values of a, b, and c

B. a * (b + c) = a + b + c for all values of a, b, and c

C. a * (b + c) = b * c for all values of a, b, and c

D. a * (b + c) = 1 for all values of a, b, and c

10. The group of integers under addition has the identity element:

A. -1

B. 0

C. 1

D. ∞

11. Consider the Set Q of rational numbers, and let * be the operation on Q defined by a*b=a+b-ab. Then 3*4 = _________.

A. 4
B. -1
C. -4
D. -5

12. The group of real numbers under multiplication has the identity element:

A. -1
B. 0
C. 1
D. ∞

13. Which of the following is a binary operation?

A. Addition of two numbers

B. Square root of a number

C. Differentiation of a function

D. Exponentiation of a number

14. The group of invertible 2x2 matrices with matrix multiplication has the identity element:

A. The zero matrix

B. The identity matrix

C. The inverse matrix

D. The diagonal matrix

15. A semigroup can have:

A. One or more than one identity elements

B. Exactly one identity element

C. Exactly two identity elements

D. No identity element

16. The monoid of natural numbers under addition has the identity element:

A. -1

B. 0

C. 1

D. ∞

17. Which of the following statements is TRUE about subgroups?

A. Every group has only one subgroup

B. Every subgroup is a group

C. Every subgroup contains only the identity element

D. Every group is a subgroup