The Basics of Proofs Spring 2010 BYU Math Department Today's - - PowerPoint PPT Presentation

The Basics of Proofs

Spring 2010

BYU Math Department

Today's Topics

1)Using definitions, theorems and laws 2)Using a counter-example 3)Proof by Contradiction 4)Proof by Induction

General Rule of Thumb

1)What do they want? 2)What do I know? 3)How can I use what I know to get what they want? When you see a problem, always ask the following questions:

Using Definitions, Theorems, and Laws

Proofs are very different from the traditional math problem. Often the problem won't tell you all the information you need. This is why it's important to know the definitions, theorems, and laws listed throughout the section, chapter, and book.

Let's try an example

First, you need some definitions.

These may seem obvious, but pretend that you've never been taught this stuff before. 1 + 1 = 2 2 + 3 = 5 Prove: 1 + 1 + 3 = 5

Let's start by asking the basic questions

1)What do they want? 2)What do I know? 3)How can I use what I know to get what they want?

Example 1, cont.

What do I know?

1 + 1 = 2 2 + 3 = 5

What do they want?

1 + 1 + 3 = 5

How do I use what I know to get what they want?

Since 1 + 1 = 2 Then 1 + 1 + 3 = 2 + 3 And we know that 2 + 3 = 5 Therefore 1 + 1 + 3 = 5. QED

=>

Quick Gear Change!

You shouldn't (and your professors won't!) use standard English to write out the steps of your proofs. We will use notation instead. => “implies that” Ǝ “there exists” Ʉ “for all” ϶ or s.t “such that”

...

“therefore” QED quod erat demonstrandum (end of proof)

Quick Gear Change!

Since 1 + 1 = 2 Then 1 + 1 + 3 = 2 + 3 And we know that 2 + 3 = 5 Therefore 1 + 1 + 3 = 5. QED

becomes

1 + 1 = 2 => 1 + 1 + 3 = 2 + 3 2 + 3 = 5 =>1 + 1 + 3 = 5. QED

So let's 'fix' our last proof

Let's try something more realistic

Suppose the following:

Jerry needs help if and only if Jerry is a rudy pants. Jerry is loony if Jerry needs help. Jerry is a rudy pants.

Prove:

Jerry is loony.

Example 2, cont.

What do I know? What do they want?

Prove that Jerry is loony

How do I use what I know to get what they want?

Jerry is a rudy pants => Jerry needs help => Jerry is loony QED Jerry needs help if and only if Jerry is a rudy pants. Jerry is loony if Jerry needs help. Jerry is a rudy pants.

Let's try something more math-like

Prove:

If f(x) is any kind of even function, then f(x) is not a one-to-one function

Example 3, cont.

How is this problem different? All of the information you need has not been given to you in the stating of the problem. They assume that you know your definitions

• f even/odd/one-to-one functions.

Quick Review! If f(x) is an even function, then f(-x) = f(x) If f(x) is a one-to-one function, then every value f(x) can be reached by one and only one value x.

What do I know? What do they want?

Prove that f(x) is not one-to-one

How do I use what I know to get what they want?

f(x) is even => f(-x) = f(x) => Ǝa,b ϶ f(a) = f(b) =>There is more than one corresponding x for every f(x) ... f(x) is not one-to-one. QED f(x) is even If f(x) is even, then f(-x) = f(x) If a function is one-to-one, then for every f(x) there is one and only one corresponding x.

Example 3, cont.

Let's try one more example

Prove:

If y = x, and y = x2 then y = 0 or 1

Any Volunteers?

What do I know? What do they want?

Prove that y = 0 or 1

How do I use what I know to get what they want?

y = x y = x2 => x2 = x => x2- x = 0 => x(x - 1) = 0 => x = 0, 1 y = x ... y = 0, 1 QED y = x y = x2 If a = b, and b = c then a = c

Example 4, cont.

Tips and Tricks for Future Problems

If you just can't seem to solve the problem... 1)Make sure you found all possible definitions, theorems, and laws that are associated with the problem. 2)Start writing something!

– Most of the time you won't see how to prove it

just by thinking about it.

3)Try to understand the proof conceptually.

– Sometimes understanding why the proof is true

from another angle opens your mind to possible solutions.

Using a Counter-example

Sometimes you'll be given a problem where you need to disprove something. The method to solving these problems is much simpler. All you need is a counter-example.

The Indian Example

I have a close friend who visited a nearby Indian reserve and insisted that all Native Americans walk in single file. I asked him how he knew that. He said that the one that he saw was walking in single file.

Learning from the Indian Example

Why was my friend's conclusion absurd? He used only one case to say something very general about Native Americans. What if he had told me that he knew that not all Native Americans walked in single file because he saw two walking side by side? His statement would be accurate, right?

Let's try another example

Disprove:

If a and b are any odd numbers, then a + b is always an odd number.

What do I know? What do they want?

Disprove that a + b is always

• dd.

How do I use what I know to get what they want?

a and b are odd numbers Let a = 1, b = 3 => a + b = 4 => 4 is even => a + b is even ... a + b is not always odd QED a and b are any odd numbers

Example 5, cont.

Don't forget!

You can only DISPROVE something with an example. It is NOT enough to just use an example to PROVE something generally.

Proof by Contradiction

Let's suppose that you could prove that something was never false. That would mean that it must always be true, right? That is the premise of proofs by contradiction. We will assume that the conclusion is false and show that it leads to a contradiction.

Let's try another example

Prove: 2

1/2 is an irrational number

Hint: if x

2 is even => x is even

What do I know? What do they want?

Show 2

1/2 is irrational

How do I use what I know to get what they want?

Example 6, cont.

An irrational number cannot be represented by an irreducible fraction such as p/q. x

2 is even => x is even

Assume 2

1/2 is rational

=> Ǝ p/q ϶ p/q = 2

1/2

=> p

2/q 2 = 2

=> p

2 = 2q 2

Ʉk, 2k is an even number => 2q

2 is an even number

=> p

2 is an even number

=> p is an even number => Ǝm, ϶ p = 2m => p

2 = 4m2

=> 4m

2 = 2q 2

=> 2m

2 = q 2

=> q

2 is even

=> q is even => p/q is irreducible ... Our assumption is always false => 2

1/2 is irrational

QED

Let's try one more example

Prove: There do not exist integers m and n such that 4m + 6n = 9.

What do I know? What do they want?

4m + 6n ≠ 9 for any m and n

How do I use what I know to get what they want?

Not much...

Example 7, cont.

Assume that Ǝ m, n s.t. 4m + 6n = 9 => 2(2m + 3n) = 9 => 2(2m + 3n) is even => an even number = 9 => which is false ... Our assumption is always false => 4m + 6n ≠ 9 for any m and n

Proofs by Induction

Let's suppose you wanted to prove that a pattern works for any possible case up to infinity. For example, how could you prove that 1 + 2 + … + n = n(n+1)/2 regardless of what n was? What if I could prove that it worked for n = 1? Is that enough?

Proofs by Induction, cont.

What if we could prove it was true for 1 and 2? Is that enough? 1 + 2 + … + n = n(n+1)/2 What if we could prove it was true for any random value as well as the next one? Is that enough? We need to prove both.

The three basic steps

• 1. Prove that it works for a base case
• 2. Assume that it works for a random value k
• 3. Prove that it works for the value k + 1

You will always use these three steps when working with induction.

Let's solve our problem

Prove: 1 + 2 + … + n = n(n+1)/2 for all n FYI: If a proof asks you to prove it for all values of n

• r similar, then induction will be the method

to use 99% of the time.

What do I know? What do they want?

1 + 2 + … + n = n(n+1)/2

How do I use what I know to get what they want?

Sounds like an induction problem.

Example 8, cont.

Show that it works for n = 1 => 1 = 1(1+1)/2 => 1 = 1(2)/2 = 1(1) = 1 TRUE Assume that the pattern works for n = k => 1 + 2 + … + k = k(k+1)/2 Prove that it works for n = k+1 => 1 + 2 + … + k + k+1 = (k+1)(k+2)/2 1 + 2 + … k = k(k+1)/2 => k(k+1)/2 + k + 1 = (k+1)(k+2)/2 => (k+1)(k/2 + 1) = (k+1)(k+2)/2 => (k+1)(k+2)/2 = (k+1)(k+2)/2 TRUE ... 1 + 2 + … + n = n(n+1)/2 QED

Let's do one more hard problem

Prove: 2 * 6 * 10 * … * (4n – 2) = (2n)!/n!

Any volunteers?!