Proofs

Proof Strategies

1. In a direct proof of a conditional statement, you assume that P is true, then use inference rules and other facts to prove that Q is true.
2. In an indirect proof of "P --> Q", you assume ~Q, then prove ~P.
3. In a direct proof of a statement P, you start with math facts that you know and try to derive P. (This is the way that we proved "14 is even" in the last class).
4. An indirect proof of a statement P is called a proof by contradiction. Here you assume P is false, and prove that this leads to a contradiction.

How to decide which proof strategy to use? This is to some extend based upon experience.

Today we are going to look at a collection of proofs, then work to understanding written arguments by recognizing what proof strategies they are mimicking.

Proof attempt one: we could try all possibilities and show that for every possible way of scheduling games, there is one day of the week with 6 games. This is a valid proof technique but not feasible for this problem.

Proof attempt two: The "best we can do, without having 6 on a day, is to have 5 on every day". That only gets to 35, so there must be 6 on some day. This is the right idea, but this is not a proof. Why not? This is hard to say, because the theorem is right, and the idea is right. But the concept of "the best we can do" is vague, and without making that concrete, all you've done is say "you can assign 35 games so that you have 5 on each day of the week.

Proof attempt three: We prove this through contradiction.

Assume, for contradiction, that the 41 games are scheduled so that no
day of the week has more than 5 games.

Then all days have less than or equal to 5 games.

The most possible games is then 7*5 = 35 games.

This contradicts that all 41 games are scheduled.

Therefore some day has at least 6 games.

This is a "paragraph" style proof.  Helps to still write out as statements.

Brief interlude. Definition of divides and relatively prime.

a | b, exists k such that ak = b.

a,b relatively prime. forall k>1, not (k | a and k | b)

Refinement of definition of rational:

a is rational <==> there exists b,c such that a = b/c and b,c are relatively prime.

Prove sqrt(2) is irrational

(if you've seen this already, try proving sqrt(3) is irrational, or that the k-th root of 2 is irrational.

Lets try a direct proof.

let a = sqrt(2)

~(exists) b,c such that a = b/c

forall b,c, a != b/c.

ARggh. what to do? what to do?

Lets now try a proof by contradiction

Suppose, for contradiction, that sqrt(2) is rational

(exists) b,c, relatively prime such that sqrt(2) = b/c

square both sides: 2 = b²/c²

2c² = b²

so b² is even (by defn of even)

so b is even (proof in last class)

so b = 2k (defn of even).

2c² = (2k)²

2c² = 4k²

c² = 2k²

c² is even defn of even

c is even proof yesterday in class

so b,c are not relatively prime

contradicts defn of rational.

You can cover all squares of a checkerboard with 2x1 dominoes.

If you remove two opposite corners of checkerboard, it is not possible to cover all squares.

P --> (Q v R)
Q --> S
R --> S
--------
P --> S

or 

(Q v R)
Q --> S
R --> S
--------
S

Now Lets move to something different. The official GRE rules are: "Discuss how well reasoned you find the following argument"

The following appeared in a magazine for the trucking industry.

"The Longhaul trucking company was concerned that its annual accident rate (the number of accidents per mile driven) was too high. It granted a significant pay increase to its drivers and increased its training standards. It also put strict limits on the number of hours per week each driver could drive. The following year, its trucks were involved in half the number of accidents as before the changes were implemented. A survey of other trucking companies found that the highest-paid drivers were the least likely to have had an accident. Therefore, trucking companies wishing to reduce their accident rate can do so simply by raising their drivers' pay and limiting the overall number of hours they drive." An excellent way to prepare for the "Analyze an Argument" task is to practice writing on some of the published Argument topics. There is no one way to practice that is best for everyone. Some prefer to start practicing without adhering to the 30-minute time limit. If you follow this approach, take all the time you need to analyze the argument. No matter which approach you take, you should

1. carefully read the argument, you might want to read it over more than once
2. identify as many of its claims, conclusions, and underlying assumptions as possible
3. think of as many alternative explanations and counterexamples as you can
4. think of what additional evidence might weaken or lend support to the claims
5. ask yourself what changes in the argument would make the reasoning more sound

   An excellent way to prepare for the "Analyze an Argument" task is to practice writing on some of the published Argument topics. There is no one way to practice that is best for everyone. Some prefer to start practicing without adhering to the 30-minute time limit. If you follow this approach, take all the time you need to analyze the argument. No matter which approach you take, you should

more examples: link