# 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 $$\Box$$.