Pure Mathematics A3: Language And Proof

13 flashcards · Mathematics · by lakesare

  1. Proposition VS variable proposition?

    propositions: P, Q, R variable propositions: P(x), Q(x)

  2. Lemma VS corollary?

    lemma - theorem that is used in the proof of other theorems. corollary - theorem that follows from another theorem by a short additional argument.

  3. Name 2 names for the P → Q connective.

    conditional implication

  4. Restate this theorem as an implication (P → Q).

    Many theorems have statements of the form: Let P. Then Q. This is an alternative way to express a theorem of the form: If P, then Q.

  5. Name 2 ways to name P ⟷ Q:

    iff equivalence

  6. Implication VS equivalence?

    implication: P → Q equivalence: P ⟷ Q

  7. What's one exception when we write "if", but mean "if and only if" in mathematics?

    In definitions.

  8. How should this statement strictly be expressed?

    P(x) → Q(x) is always the universal (∀x)(P(x) → Q(x)) statement. So implications where the hypothesis and the conclusion are variable propositions are in fact universal statements where the universal quantifier is omitted.

  9. What book did Alfred North Whitehead and Bertrand Russel wrote? Bertrand Russel Alfred North Whitehead

    Principia Mathematica (1910–1913)

  10. What, in logical sense, are we asked to show when we're asked to solve the equation?

    Whenever we solve an equation, an implication and its converse must both be proved - we need to prove an equivalence.

  11. What's wrong with this proof (last 2 lines)?

  12. Name a fast way to determine whether m is prime. We could simply check whether m is divisible by each of the integers between 2 and m − 1 inclusive, but this involves a large amount of calculation even for fairly small integers m. We can significantly reduce the amount of calculation needed by using the fast method, which holds for any integer m ≥ 2.

    m is prime if it's not divisible by any primes ≤ √m.

  13. Name 2 ways to prove an equality.

    Transform the LHS and RHS independently. Transform one side of the equation until it's seen to be the same as the other side.