MAT 3749Introduction to Analysis

Section 0.1 Part I

Methods of Proof I

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Preview of Reviews

Set up common notations. Direct Proof Counterexamples Indirect Proofs - Contrapositive

Background: Common Symbols and Set Notations

Integers This is an example of a Set: a collection

of distinct unordered objects. Members of a set are called elements. 2,3 are elements of , we use the

notations

{ , 3, 2, 1,0,1,2,3, }

2 , 3

2,3

Background: Common Symbols

Implication Example

If 2 then 3 5

2 3 5

x x

x x

Goals

We will look at how to prove or disprove Theorems of the following type:

Direct Proofs Indirect proofs

If ( ), then ( ).statements statements

Theorems

Example:

2 If is odd, then is also odd.m m

Theorems

Example:

Underlying assumption:

2 If is odd, then is also odd.m m

Hypothesis Conclusion

m

Example 12 If is odd, then is also odd.m m

Analysis Proof

Note

Keep your analysis for your HW Do not type or submit the analysis

Direct Proof2 If is odd, then is also odd.m m

Direct Proof of If-then Theorem• Restate the hypothesis of the result. • Restate the conclusion of the result.• Unravel the definitions, working forward from the beginning of the proof and backward from the end of the proof.• Figure out what you know and what you need. Try to forge a link between the two halves of your argument.

Example 2

If is odd and is even, then is odd.m n m n

Analysis proof

Counterexamples

To disprove

we simply need to find one number x in the domain of discourse that makes false.

Such a value of x is called a counterexample

x P x

P x

Example 3

, 2 1 is primenn N

Analysis The statement is false

Example 4 If 3 2 is odd, then is also odd.n n

Indirect Proof: Contrapositive

To prove

we can prove the equivalent statement in contrapositive form:

or

If 3 2 is odd, then is also odd.n n

If is odd, then 3 2 is not no d.t odn n

If is , theven en 3 2 is en.evn n

Rationale

Why? If 3 2 is odd, then is also odd.n n

If is odd, then 3 2 is not no d.t odn n

Background: Negation

Statement: n is odd

Negation of the statement: n is not odd

Or: n is even

Background: Negation

Notations

Note:

P: is odd

~P: is not odd

n

n

Some text use P

Contrapositive

The contrapositive form of

is

If then P Q

If ~ then ~Q P

Example 4

Analysis Proof: We prove the contrapositive:

If 3 2 is odd, then is also odd.n n

If is even, then 3 2 is even.n n

Contrapositive

Analysis Proof by Contrapositive of If-then Theorem• Restate the statement in its equivalent contrapositive form.• Use direct proof on the contrapositive form.•State the origin statement as the conclusion.

If 3 2 is odd, then is also odd.n n

Classwork

Very fun to do. Keep your voices down…you do not

want to spoil the fun for the other groups.