Equivalence Relations and Partitions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Sorting sets with equivalence relations

Mathematics · Mathematical Thinking · ages 18-19
Name ______________________   Date ____________

  1. Which properties make equality on a set an equivalence relation?

    • Reflexive, symmetric and transitive
    • Reflexive only
    • Symmetric only
    • Transitive only

  2. Which properties make equality on a set an equivalence relation?

    • Reflexive only
    • Symmetric only
    • Reflexive, symmetric and transitive

  3. How many distinct equivalence classes does congruence modulo 5 have?

    Answer: ______________

  4. What does reflexivity require of a relation?

    • Every element relates to itself
    • Each pair appears both ways
    • Chains link through intermediates

  5. If the classes of a and b share an element c, what follows?

    • The classes coincide, since c links each member to the other
    • They overlap partly but stay different
    • One of the classes must be empty

  6. If the classes [a] and [b] share an element c, what follows?

    • [a] = [b], since c links every member of one class to the other
    • They overlap partly but stay different
    • One of the classes must be empty
    • Nothing follows

  7. For integers with a ~ b meaning a < b, which property fails?

    • Reflexivity, since a < a never holds
    • Symmetry only, while reflexivity holds
    • None: it is an equivalence relation
    • Transitivity

  8. For integers with a related to b meaning a is less than b, which property fails?

    • Transitivity
    • Reflexivity, since a is never less than itself
    • None, it is an equivalence relation

  9. Lee says the classes of 2 and 5 modulo 3 coincide, since 5 minus 2 equals 3.

    Circle one:   True   False

  10. A relation is reflexive and symmetric but chains do not always link through. What is it?

    • A partial overlap of two classes
    • An equivalence relation
    • A near miss, since transitivity fails
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Sorting sets with equivalence relations W1-mt_wgEupYwEUF-s1

  1. Reflexive, symmetric and transitive · Equality satisfies all three: a = a, a = b gives b = a, and chains link up.
  2. Reflexive, symmetric and transitive · Equality meets all three: a equals a, pairs reverse, and chains link up.
  3. 5 · The possible remainders are 0, 1, 2, 3 and 4: five classes.
  4. Every element relates to itself · Reflexivity is the self check: no element may skip relating to itself.
  5. The classes coincide, since c links each member to the other · Symmetry and transitivity move every member across c, so each class sits inside the other.
  6. [a] = [b], since c links every member of one class to the other · From x ~ c and c ~ y, transitivity and symmetry give x ~ y, so each class sits inside the other.
  7. Reflexivity, since a < a never holds · Reflexivity fails at once since no integer is less than itself; transitivity actually holds.
  8. Reflexivity, since a is never less than itself · The self check fails at once, while transitivity actually holds for less than.
  9. True · 3 divides the difference, so 5 lies in the class of 2 and the classes match.
  10. A near miss, since transitivity fails · All three properties are required, so a missing one means it is not an equivalence relation.
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