1.1 Intro to Proofs

Modus Ponens

Let \(P\) be a proposition, \(Q\) be another proposition.

  1. What is a proposition of the form \(\text{IF } P, \text{ THEN } Q \) called?

    Exercise 1
    \(\text{IF } P, \text{ THEN } Q \) is the general form of an implication and is often written as \(P \text{ IMPLIES } Q\). Thus, given specific \(P\) and \(Q\), \(P \text{ IMPLIES } Q\) is itself a proposition and can be either true or false.
  2. A fundamental inference rule says:

    \(\dfrac{P,\;\; P \text{ IMPLIES } Q}{Q}\).
    1. What is this inference rule called?

      Exercise 2
    2. What is the statement above the line called?

      Exercise 3
    3. What is the statement below the line called?

      Exercise 4
    Review Chapter 1.4.1.
  3. Proving a proposition's contrapositive is as good as (and sometimes easier than) proving the proposition itself. Which of the following is logically equivalent to the contrapositive of \(P \text{ IMPLIES } Q\)?

    Exercise 5
    Draw a Venn diagram with \(P\) inside of \(Q\).
  4. At the end of a proof, it is customary to write down either the delimiter _____ or the symbol _____.

    Exercise 6
    A proof should begin with "Proof by ..." and end with "QED" or \(\Box \).