MAT 2720Discrete Mathematics

Section 2.1 Mathematical Systems,

Direct proofs, and Counterexamples

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Preview Set up common notations. Direct Proof Counterexamples

Implication Implication Example

If 2 then 3 5

2 3 5x xx 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

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

Proof: 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

Example 3

Analysis Proof

A B A B

A B

A B

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 4, 2 1 is primenn Z

Analysis The statement is false

MAT 2720Discrete Mathematics

Section 2.2 More Methods of Proof

Part I

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Goals Indirect Proofs

• Contrapositive• Contradiction• Proof by Contrapositive is considered as a

special case of proof by contradiction• Proof by cases

Example 1 If 3 2 is odd, then is also odd.n n

Indirect Proof: ContrapositiveTo 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

RationaleWhy?

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 (1.2)Statement: n is oddNegation of the statement: n is not odd Or: n is even

Background: Negation (1.2)Notations

Note:

P: is odd~P: is not odd

nn

The text uses P

Contrapositive (1.3) The contrapositive form of

is

If then P Q

If ~ then ~Q P

Example 1

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