Statements, Quantifiers and Negation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Quantifiers, negation and conditionals

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

  1. What is the negation of: some birds cannot fly?

    • Every bird can fly
    • No bird can fly at all, ever
    • Some birds can fly sometimes

  2. Which statement is the negation of (A and B)?

    • (not A) and (not B)
    • A or B
    • (not A) or (not B)

  3. Claim: if it rains, the match is cancelled. Which sentence is the contrapositive?

    • If the match is cancelled, it rained
    • If the match is not cancelled, it did not rain
    • If it does not rain, then the match is certainly not cancelled

  4. What is the negation of: some birds cannot fly?

    • Every bird can fly
    • No bird can fly
    • Some birds can fly
    • Every bird cannot fly

  5. Leo says the converse of a true conditional is always true. Take: if x is a multiple of 4 then x is even, and note 6 is even but no multiple of 4. Is Leo right?

    Circle one:   True   False

  6. Zoe says swapping two for every quantifiers never changes what a statement claims. Is Zoe right?

    Circle one:   True   False

  7. What is the negation of: for every x there is a y with y greater than x?

    • There is an x such that for every y, y is less than or equal to x
    • For every x there is a y with y less than or equal to x
    • There is an x and a y with y greater than x
    • For every y there is an x with y greater than x

  8. What is the negation of: for every x there is a y with y greater than x?

    • For every x there is a y with y less than or equal to x, flipping only the inner claim
    • There is an x and a y with y greater than x, flipping nothing at all
    • There is an x such that for every y, y is less than or equal to x

  9. Zoe says swapping two for every quantifiers never changes what a statement claims. Is Zoe right?

    Circle one:   True   False

  10. Negate this claim: there is a city where every bus is late.

    • For every city there is a bus that is not late
    • There is a city where some particular bus is never late
    • For every city every bus is always late
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Answer key

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

Quantifiers, negation and conditionals W1-mt_S5NvyPxEZ_-s1

  1. Every bird can fly · Not some becomes every, and cannot fly becomes can fly.
  2. (not A) or (not B) · De Morgan turns not-and into or of the negated parts.
  3. If the match is not cancelled, it did not rain · The contrapositive negates both halves and swaps them.
  4. Every bird can fly · Not some becomes every, and cannot fly becomes can fly, giving every bird can fly.
  5. False · The conditional is true but its converse fails at 6, so Leo is wrong.
  6. True · For every x, for every y says the same as for every y, for every x, since both demand the claim for all pairs.
  7. There is an x such that for every y, y is less than or equal to x · Negation flips each quantifier and the inner claim: for every becomes there is, there is becomes for every, and greater than becomes less than or equal to.
  8. There is an x such that for every y, y is less than or equal to x · Flip each quantifier and the inner claim: for every becomes there is, there is becomes for every, greater than becomes less than or equal to.
  9. True · Both orders demand the claim for every pair, so swapping changes nothing and Zoe is right.
  10. For every city there is a bus that is not late · Flip the outer quantifier to for every, the inner to there is, and late to not late.
Worksheet · LightMySky