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. On S = {u, v}, a relation contains (u, u), (v, v) and (u, v), but not (v, u). Which pair shows that symmetry fails?

    • (u, u), because it relates u to itself
    • (v, v), because it relates v to itself
    • (u, v), because its reverse is missing
    • (v, u), because it belongs to the relation

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

    • Reflexive and symmetric, without transitivity
    • Symmetric and transitive, without reflexivity
    • 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?

    • They coincide as sets
    • They overlap but have different members
    • One class has no elements despite containing c

  6. On {a, b, c}, an equivalence relation consists of (a, a), (a, b), (b, a), (b, b) and (c, c). Which partition does it produce?

    • {{a}, {b, c}}
    • {{a, b, c}}
    • {{a}, {b}, {c}}
    • {{a, b}, {c}}

  7. On {1, 2, 3}, a relation has all self-pairs and the pairs (1, 2), (2, 1), (2, 3), (3, 2). Which chain disproves transitivity?

    • 1 ~ 2 and 2 ~ 1, with 1 ~ 1 present
    • 1 ~ 2 and 2 ~ 3, with 1 ~ 3 absent
    • 2 ~ 3 and 3 ~ 2, with 2 ~ 2 present
    • 3 ~ 2 and 2 ~ 2, with 3 ~ 2 present

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

    • Transitivity: a < b < c never gives a < c
    • Reflexivity: a < a never holds
    • All three properties hold for this 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?

    • An equivalence relation because two properties suffice
    • An equivalence relation because symmetry implies transitivity
    • Not an equivalence relation: 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. (u, v), because its reverse is missing · Symmetry requires the reverse of every included pair. The pair (u, v) is included, but (v, u) is missing.
  2. Reflexive, symmetric and transitive · Equality satisfies all three properties: a equals a, equality pairs can be reversed, and equality chains are transitive.
  3. 5 · The possible remainders are 0, 1, 2, 3 and 4: five classes.
  4. Every element relates to itself · Reflexivity requires every element to relate to itself.
  5. They coincide as sets · Using the shared element c, symmetry and transitivity show that each class is contained in the other.
  6. {{a, b}, {c}} · The elements related to a are a and b. The only element related to c is c.
  7. 1 ~ 2 and 2 ~ 3, with 1 ~ 3 absent · The chain 1 ~ 2 ~ 3 would require 1 ~ 3. That pair is absent.
  8. Reflexivity: a < a never holds · 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. Not an equivalence relation: transitivity fails · All three properties are required, so a missing one means it is not an equivalence relation.
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