Proof 101

Proof by contradiction usually...

Proof by contradiction is often used when you wish to prove the impossibility of something. You assume it is possible, and then reach a contradiction. In the examples below we use this idea to prove the impossibility of certain kinds of solutions to some equations.

Proof by Contradiction

If P, then Q prove by showing that If P, then NOT Q is FALSE (that you will reach some kind of contradiction)

Note for when to use Proof by Contradiction

If you're trying to prove the converse of an ALREADY EXISTING PROOF (so you know this statement is a converse) say the statement is if P, then Q (and you know this is a converse from an existing truth If Q, then P- b/c If Q then P is already a theorem/ proposition/ definition etc) then to prove if P, then Q, USE PROOF BY CONTRADICTION If P, then NOT Q

Example of Mathematical Induction

Initial Step. When n = 0, the formula gives us (1 - 1/22n)/2 = (1 - 1/2)/2 = 1/4 = a0. So the closed form formula ives us the correct answer when n = 0. Inductive Step. Our inductive assumption is: Assume there is a k, greater than or equal to zero, such that ak = (1 - 1/22k)/2. We must prove the formula is true for n = k+1. First we appeal to the recurrsive definition of ak+1 = 2 ak(1-ak). Next, we invoke the inductive assumption, for this k, to get ak+1 = 2 (1 - 1/22k)/2 (1 - (1 - 1/22k)/2) = (1 - 1/22k)(1 + 1/22k)/2 = (1 - 1/22k+1)/2. This completes the inductive step.

Important Negations (Given this as Q, what is Not Q) Q: A or B

Not Q: Not A AND Not B

Q: A and B

Not Q: Not A OR Not B

if Q: at least one rational solution

Not Q: no rational solutions

if Q: all rational solutions

Not Q: there exists at least one not rational solution

ex of proof by contradiction

There are no rational number solutions to the equation x3 + x + 1 = 0. in P Q form, we have if it is a solution to x3 +x+1=0 (P) , the solution cannot be a rational number solution (Q) so using proof by contradiction, assume there is AT LEAST ONE rational number solution

Direct Proof

To prove -if P, then Q or just P, then direct proof comes from starting @ implications of P, and then getting to P start w/ assumptions coming from if P....... then go directly to show that Q will be true

Mathematical Induction

Usually used for long series Initial step: prove the first item in the series is true P(1) is true Inductive Step. Here we must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true." Thus, we assume there is a k such that 1 + 2 + ... + k = k (k+1)/2. (We call this the inductive assumption.) We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.

Constructive Proofs (Strategy 2)

construct a number using components/ not the actual number ex. There is a rational number that lies strictly between the square root of 10100 and the square root of 10100+1. Proof. The square root of 10 100 is 10 50. After a little bit of trial and error, we let x = 10 50 + 10 -51, which is clearly a rational number bigger than the squre root of 10 100. To prove that x is less than the square root of 10100+1, we compute x2 = (10 50 + 10 -51)2 = 10 100 + (2) 10 -1 + 10 -102 which is clearly less than 10100+1.

Existential Proofs (Strategy 3)

don't have to construct a number, sometimes by a theory, you know something exists (ex. nontrivial solution exists for Ax=0 if A is nonsingular)

Contradiction (Strategy 4)

great way to use when proving the converse of if P, then Q (so if Q, then P) is false

Definitions (strategy 1)

unwind the definitions- underline ALL definitions in the P and Q and see what these definitions imply (ex. nonsingular definition has 3 parts to it)

Helpful strategies for Direct Proof

write P out in terms of something else (ex. for the midpoint of line proof, we wrote the midpoint in terms of the two vectors x and y)..... try and find a "something else" that has a property you can use for each step, find a preposition/ theorem to use to advance you further

Proof by Contrapositive

you're trying to prove : If P, then Q Instead prove if NOT Q, then NOT P So instead of end goal is to prove Q, your end goal is to start with not q and prove not p is true (you have an end goal!)