Essential Mathematics: Language And Proof

Mathematical language and proof.

8 flashcards · Mathematics · by lakesare

  1. Conjecture VS theorem?

    Conjecture is NOT yet proved. Mathematicians use the word conjecture to describe a statement that seems likely to be true but has not been proved to be true. One of the most famous conjectures is the twin prime conjecture, which is the following statement: "There are infinitely many prime numbers p such that p + 2 is also a prime number".

  2. Lemma VS theorem?

    The word lemma is usually used to describe a true statement that is not important enough to be called a theorem. It is often used for a statement that can be used to prove a more important statement.

  3. Give an example of a variable proposition. Mathematical statements, aka propositions: 2 + 2 = 3.

    x is greater than 0. n is a multiple of 3. For example, statement 1 above is true when x = 2 but false when x = −1. It is sometimes convenient to use the notation P(x) or P(n) to denote a variable proposition that depends on a variable x or n, respectively In particular, a mathematical statement that is either true or false depending on the values of one or more variables, such as x or n, is called a variable proposition

  4. Write down the converse of the following implication: Every integer that is a multiple of 4 is also a multiple of 8.

    Every integer that is a multiple of 8 is also a multiple of 4. The converse of the implication 'if P, then Q' is the implication 'if Q, then P'. In other words, the converse of the implication P ⇒ Q is the implication Q ⇒ P.

  5. [Rewriting implications] Rewrite the implication in the form '... ⇒ ...'. x ≥ 2 only if x2 ≥ 4.

    x ≥ 2 ⇒ x2 ≥ 4.

  6. Name 2 symbols representing iff.

  7. Why is this proof wrong?

    It takes the assertion, and if-then-s from it! Instead we should use iff-s to prove it. Here is a correct proof !!! (uses if and only if-s)

  8. How would you go about proving this?

    Use a proof by contraposition. If you want to prove a particular implication, and you are finding it hard to use a direct method of proof, then you may find it easier to show that the contrapositive is true.