CS0441 lecture notes (review of 1.6, continuation of 1.7)

Direct proof refresher example

Given

• If I work all night on this homework, then I can answer all the exercises.
• If I answer all the exercise, I will understand the material.

Prove

• If I work all night on the homework, then I will understand the material

Let

• P = I work all night on this homework
• ?

	Proposition	Justification	Applied to
1	P	Assume
?

Indirect proof refresher example

Redo example above

Indirect Proof and Proof by cases

This proof uses a special technique, called proof by cases, which decomposes a proposition of the form AVB into cases A and B, and proves the same conclusion in both cases, then recombines them.

Example: Given

· If I ate spicy food, I will have strange dreams

· If it thunders while I sleep, I will have strange dreams

Prove

· If I do not have strange dreams, then I didn’t eat spicy food and it didn’t thunder last night.

Let

· P=I ate spicy food

· R=it thundered last night

· Q=?

	Proposition	Justification	Applied to
1	~(~P^~R)	Assumed
2	P vR	Demorgans & double negation
3	P	First case	2
4	P->Q	Given
5	Q	Modeus ponens	3,4
6	R	Second case	2
7	?	?
8	?	?	?
9	Q	Proof by cases	2, 5, 8

Existence proofs: Constructive

If the proposition to be proved has an existential quantifier in it, then its proof is called an existence proof. One way to prove such a proposition is to construct a constant that satisfies the proposition when it is substituted for the existentially quantified variable.

Example: Prove “For every n, there is an integer divisible by more than n different prime numbers.” That is, For all n, there exists k such that k is divisible by more than n different prime numbers

	Proposition	Justification	Applied to
1	Let {3, 5, 7, …, pn+1) be the first n+1 prime numbers.	There are infinitely many prime numbers
2	Let k = 3*5*…*pn+1	You can multiply any number of numbers together	1
3	k is divisible by pi for all i between 1 and n+1	Because pi is a factor of k	2
4	k is divisible by n+1 different prime numbers	counting	3

Existence proofs: Non-constructive

If an existence proof does not actually construct the constant that makes the proposition true, then the proof is called a non-constructive existence proof.