Sets and Functions in the Language of Proof · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Proving with sets and functions

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

  1. Kim says that under f(x) equal to x squared, the preimage of {9} always has exactly one element. Is Kim right?

    Circle one:   True   False

  2. Let f map real numbers to real numbers by f(x) equal to x squared. Which fact shows f is not injective?

    • f(2) equals f(-2), both equal to 4
    • f(0) equals 0
    • f(3) equals 9
    • f(1) equals 1

  3. To prove that two sets A and B are equal by double inclusion, which two containments do you show?

    • B inside A, proved twice
    • A inside B and B inside A
    • That both sets are empty

  4. With f(x) equal to x squared, what is the image of the set containing minus 2 and 3?

    • The set containing minus 2 and 3
    • The set containing 2 and 3
    • The set containing 4 and 9

  5. Kim says that under f(x) equal to x squared, the preimage of the set containing 9 holds exactly one element.

    Circle one:   True   False

  6. With f(x) equal to x squared, what is the image of the set {-2, 3} and the preimage of the set {4}?

    • Image {4, 9}; preimage {-2, 2}
    • Image {-2, 3}; preimage {4}
    • Image {4, 9}; preimage {2}
    • Image {2, 3}; preimage {-4, 4}

  7. To prove that A union (B intersect C) equals (A union B) intersect (A union C) by double inclusion, what is the correct first half?

    • Take an arbitrary x in the left hand side and show it lies in the right hand side
    • List every element of A, B and C
    • Assume both sides are empty
    • Prove the converse statement instead

  8. You have shown that every element of the left side lies on the right side. What still remains in a double inclusion proof?

    • The reverse chase, from the right side back to the left
    • A longer Venn picture
    • A check that both sides are nonempty

  9. Suppose f and g are injective and f(g(a)) equals f(g(b)). What follows?

    • Nothing follows without more checks
    • The composition must be constant
    • a equals b, since injectivity applies at both layers

  10. Lee proved that the left side sits inside the right side and stopped, then claimed the sets are equal. What is wrong?

    • The reverse containment is still unproved
    • Nothing is wrong, half the proof is enough
    • The two circle Venn picture is required
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Answer key

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

Proving with sets and functions W1-mt_XAcHX_3DVz-s1

  1. False · Both -3 and 3 square to 9, so the preimage holds two elements, not one.
  2. f(2) equals f(-2), both equal to 4 · Injective maps never send two different inputs to one output, but 2 and -2 both land on 4.
  3. A inside B and B inside A · Mutual containment means the same elements, which is exactly what equality of sets means.
  4. The set containing 4 and 9 · Push each input forward through squaring: minus 2 gives 4 and 3 gives 9.
  5. False · Both 3 and minus 3 square to 9, so the preimage holds two elements.
  6. Image {4, 9}; preimage {-2, 2} · Squaring gives f(-2) equal to 4 and f(3) equal to 9, while the inputs landing on 4 are -2 and 2.
  7. Take an arbitrary x in the left hand side and show it lies in the right hand side · Double inclusion proves equality by chasing an arbitrary element each way, starting left to right.
  8. The reverse chase, from the right side back to the left · One direction is only half the job, since equality needs containment both ways.
  9. a equals b, since injectivity applies at both layers · The outer chase gives g(a) equals g(b), then the inner chase gives a equals b.
  10. The reverse containment is still unproved · Equality needs both directions, and Lee only chased one of them.
Worksheet · LightMySky