DG1-DG4 Binary operations, commutativity, associativity and Cayley tables Quiz: AQA Further Maths, Unit 5

The 20 questions

1. What is a binary operation on a set?

   • A rule that combines any two elements of the set to give an element of the same set
   • A rule that takes one element of the set and returns a number
   • A rule that combines two sets to give a third set
   • A rule that combines two elements, but may give an element outside the set

2. What does it mean for a binary operation * to be commutative?

   • a * (b * c) = (a * b) * c for all elements a, b and c in the set
   • a * a = a for every element a in the set
   • a * b = b * a for all elements a and b in the set
   • a * b = a for every element b in the set

3. What does it mean for a binary operation * to be associative?

   • For each a and b there is exactly one element c with a * b = c
   • There is an element e with a * e = a for every a in the set
   • a * b = b * a for all elements a and b in the set
   • (a * b) * c = a * (b * c) for all elements a, b and c in the set

4. What is a Cayley table for a set under a binary operation?

   • A table listing only the identity and inverse of each element
   • A table listing the result of the operation for every ordered pair of elements of the set
   • A table listing the powers of a single element of the set
   • A table counting the solutions of a * x = b for each b

5. In Z5 under addition modulo 5, what is 3 + 4?

   • 2
   • 4
   • 1
   • 7

6. Is subtraction of integers commutative?

   • No, since subtraction is not defined on integers
   • No, since 5 - 2 = 3 but 2 - 5 = -3
   • Yes, since subtraction of integers is always commutative
   • Yes, since 5 - 2 = 2 - 5 for all integers

7. On the integers, the operation a * b = a + 2b is defined. Is it commutative?

   • Yes, since both 1 * 0 and 0 * 1 equal 1
   • Yes, since addition is commutative
   • No, since 1 * 0 = 1 but 0 * 1 = 2
   • No, since 1 * 0 = 0 but 0 * 1 = 1

8. On the real numbers, a * b = (a + b)/2. Is this operation associative?

   • Yes, since the operation is an average and averages are associative
   • No, since with a = 1, b = 0 and c = 0 the two groupings give 1/4 and 1/2
   • No, since the operation is not defined when one of the numbers is zero
   • Yes, because the operation is commutative

9. Which statement about multiplication of 2 × 2 matrices is correct?

   • It is commutative but not associative
   • It is associative but not, in general, commutative
   • It is both commutative and associative
   • It is neither commutative nor associative

10. A Cayley table is constructed for a set with 5 elements. How many entries does the table have?

    • 20
    • 5
    • 10
    • 25

11. A Cayley table is symmetric about its leading diagonal. What does this show?

    • Every element is its own inverse
    • The operation is associative
    • The operation is commutative
    • The set has exactly one identity element

12. In Z4 under multiplication modulo 4, what is 2 × 3?

    • 0
    • 2
    • 1
    • 6

13. In the Cayley table of addition modulo 4 on {0, 1, 2, 3}, what is the entry in the row for 2 and the column for 3?

    • 3
    • 0
    • 1
    • 5

14. Is the set {1, 2, 3} closed under multiplication modulo 4?

    • Yes, since 2 × 2 = 4 is in the set
    • No, since 2 × 2 = 4, which is 0 modulo 4 and is not in {1, 2, 3}
    • No, since 1 × 3 = 3 is not in the set
    • Yes, since every product of two elements lies in {1, 2, 3}

15. Which operation on the integers is not commutative?

    • The operation giving the larger of a and b
    • Multiplication
    • Subtraction, since 5 - 2 = 3 but 2 - 5 = -3
    • Addition

16. Does the operation a - b on the integers satisfy associativity?

    • Yes, since 1 - 2 = 2 - 1 implies associativity
    • Yes, since subtraction is associative
    • No, since (1 - 2) - 3 = -4 but 1 - (2 - 3) = 2
    • No, since subtraction is not defined for three numbers

17. A commutative binary operation is defined on a set with 3 elements. How many distinct entries must be computed in its Cayley table?

    • 6
    • 12
    • 3
    • 9

18. On the set {a, b}, a binary operation is defined by a * a = b, a * b = a, b * a = b and b * b = a. Is it commutative?

    • Yes, since every element appears once in each row of the table
    • Yes, since the operation gives the same output for each ordered pair
    • Yes, since a * a differs from b * b
    • No, since a * b = a but b * a = b

19. Which set and operation give a closed and commutative binary operation on the whole set?

    • Division on the non-zero integers
    • Addition on the integers
    • Division on the integers
    • Subtraction on the integers

20. On the set {1, 2, 3, 4}, a * b is defined as the larger of a and b. Which statement is correct?

    • The operation is commutative but not associative
    • The operation is neither commutative nor associative
    • The operation is associative but not commutative
    • The operation is both commutative and associative

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