Binary Operations - MCQs Quiz

1. What is an abstract group essentially a study of?

A) A set with an operation defined on it
B) A historical mathematical concept
C) Binary operations on numbers
D) Algebraic equations only

Correct Answer: A) A set with an operation defined on it

2. What is a binary operation ‘o’ on a set G?

A) A mapping from G to G × G
B) A mapping from G × G to G
C) A set of elements with no operations
D) A function from G to a different set

Correct Answer: B) A mapping from G × G to G

3. Which of the following is true for a commutative binary operation?

A) a o b = b o a for every pair of elements a, b ∈ G
B) a o b ≠ b o a for some pairs of elements
C) a o (b o c) = (a o b) o c
D) The operation distributes over addition

Correct Answer: A) a o b = b o a for every pair of elements a, b ∈ G

4. What does an associative operation require?

A) a o (b o c) = (a o b) o c for all a, b, c ∈ G
B) a o b = b o a
C) a o (b o c) = (a o b) o (a o c)
D) (b o c) o a = (b o a) o (c o a)

Correct Answer: A) a o (b o c) = (a o b) o c for all a, b, c ∈ G

5. When is a binary operation said to be left distributive with respect to another operation?

A) a o' (b o c) = (a o' b) o (a o' c) for all a, b, c ∈ G
B) (b o c) o' a = (b o' a) o (c o' a)
C) a o b = b o a
D) a o (b o c) = (a o b) o c

Correct Answer: A) a o' (b o c) = (a o' b) o (a o' c) for all a, b, c ∈ G

6. What is the identity element with respect to a binary operation ‘o’?

A) An element e ∈ G such that a o e = a = e o a for all a ∈ G
B) An element that does not affect the result of the operation
C) An element that can only be used with addition
D) An element that only works for multiplication

Correct Answer: A) An element e ∈ G such that a o e = a = e o a for all a ∈ G

7. Which of the following is NOT an example of an algebraic structure?

A) (R, +, .)
B) (N, +)
C) (1, -)
D) (G, o) where o is not defined

Correct Answer: D) (G, o) where o is not defined

8. In the set of real numbers R, what is the additive identity?

A) 0
B) 1
C) -1
D) Any natural number

Correct Answer: A) 0

9. What is the inverse of an integer a with respect to ordinary addition?

A) -a
B) a
C) 1/a
D) a + 1

Correct Answer: A) -a

10. For the set Q of rational numbers, if the binary operation ‘o’ is defined by a o b = a + b - ab, what property does ‘o’ exhibit?

A) Commutative and associative
B) Only commutative
C) Only associative
D) Neither commutative nor associative

Correct Answer: A) Commutative and associative

11. Which of the following sets is closed under the binary operations of union and intersection?

A) {A, B, C, D} where A = {a}, B = {b}, C = {a, b}, D = {a, b, c}
B) The set of integers
C) The set of real numbers
D) The set of natural numbers

Correct Answer: A) {A, B, C, D} where A = {a}, B = {b}, C = {a, b}, D = {a, b, c}

12. Which operation is defined as left distributive with respect to another operation?

A) a o' (b o c) = (a o' b) o (a o' c)
B) (b o c) o' a = (b o' a) o (c o' a)
C) a o (b o c) = (a o b) o c
D) a o b = b o a

Correct Answer: A) a o' (b o c) = (a o' b) o (a o' c)

13. What is an example of an identity element with respect to multiplication in the set of natural numbers N?

A) 1
B) 0
C) -1
D) 2

Correct Answer: A) 1

14. What does an algebraic structure (G, o) require?

A) A non-empty set G and at least one binary operation defined on it
B) A set with multiple binary operations
C) Only one operation with no set requirements
D) A set with operations not necessarily defined

Correct Answer: A) A non-empty set G and at least one binary operation defined on it

15. Which property must a binary operation satisfy to be considered distributive with respect to another operation?

A) Left distributive and/or right distributive
B) Commutative only
C) Associative only
D) Identity element included

Correct Answer: A) Left distributive and/or right distributive

16. For the set of integers, which is the multiplicative identity?

A) 1
B) 0
C) -1
D) 2

Correct Answer: A) 1

17. What is true about the inverse of a non-zero rational number with respect to multiplication?

A) The inverse is 1/a
B) The inverse is -a
C) The inverse is a
D) The inverse is a + 1

Correct Answer: A) The inverse is 1/a

18. What is required for a binary operation to be distributive with respect to another operation?

A) It must satisfy either left or right distributive property
B) It must be associative
C) It must be commutative
D) It must have an identity element

Correct Answer: A) It must satisfy either left or right distributive property

19. In the context of algebraic structures, what does (R, +, .) represent?

A) An algebraic structure with addition and multiplication
B) A structure with only addition
C) A structure with only multiplication
D) A non-mathematical set

Correct Answer: A) An algebraic structure with addition and multiplication

20. Which property is satisfied if a binary operation ‘o’ is defined on a set G and for all a, b, c ∈ G, a o (b o c) = (a o b) o c holds true?

A) Associative property
B) Commutative property
C) Distributive property
D) Identity property

Correct Answer: A) Associative property

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3.1 - Binary Operation - MCQs

Interactive MCQs Quiz