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

The 20 questions

1. Which property ensures a binary operation's output always lies within the original set?

   • Closure
   • Associativity
   • Commutativity
   • Distributivity

2. Which algebraic condition defines a commutative binary operation on a set?

   • (a * b) * c = a * (b * c)
   • a * e = a
   • a * b = b * a
   • a * b = e

3. Which algebraic identity defines an associative binary operation on a set?

   • a * b = e
   • a * e = a
   • a * b = b * a
   • (a * b) * c = a * (b * c)

4. What type of diagram displays all possible results of a binary operation on a finite set?

   • Truth table
   • Cayley table
   • Decision tree
   • Venn diagram

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

   • 2
   • 4
   • 1
   • 7

6. Is the binary operation of subtraction commutative on the set of integers?

   • Only for zero
   • No
   • Yes
   • Only for positives

7. For integers, is a * b = a + 2b a commutative binary operation?

   • Only for zero
   • Yes
   • No
   • Only for positive numbers

8. For real numbers, is a * b = (a + b) / 2 an associative binary operation?

   • Yes
   • No
   • Only for integers
   • Only for equal numbers

9. Which algebraic properties apply to multiplication of 2 x 2 matrices?

   • Commutative and associative
   • Associative, not commutative
   • Commutative, not associative
   • 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. What property does a Cayley table symmetric about its main diagonal demonstrate?

    • Closure
    • Associativity
    • Commutativity
    • Existence of identity

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?

    • Undefined
    • No
    • Finite
    • Yes

15. Which of these standard binary operations on integers is not commutative?

    • Maximum
    • Multiplication
    • Subtraction
    • Addition

16. Is subtraction associative on the set of integers?

    • Always
    • Yes
    • No
    • Sometimes

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 {a, b}, if a * b = a and b * a = b, is * commutative?

    • Yes
    • Only on element b
    • Only on element a
    • No

19. Which operation on the set of integers is both closed and commutative?

    • Exponentiation
    • Addition
    • Subtraction
    • Division

20. Which properties hold for the operation a * b = max(a, b) on {1, 2, 3, 4}?

    • Commutative only
    • Neither property
    • Associative only
    • Both commutative and associative

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