Methods of Proof

CS 30 : Discrete Mathematics for Computer Science First Semester, AY 2012-2013

https://sites.google.com/a/dcs.upd.edu.ph/nhsh_classes/cs30

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Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Mathematical Logic

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1

Methods of Proof Direct Proofs Indirect Proofs

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

The Hatter said, `Why is a raven like a writing-desk?' `Come, we shall have some fun now!' thought Alice. `I'm glad they've begun asking riddles. I believe I can guess that,' she added aloud. `Do you mean that you think you can nd out the answer to it?' said the March Hare. `Exactly so,' said Alice. `Then you should say what you mean,' the March Hare went on. `I do,' Alice hastily replied; `at least at least I mean what I say that's the same thing, you know.' `Not the same thing a bit!' said the Hatter. `You might just as well say that "I see what I eat" is the same thing as "I eat what I see"!' `You might just as well say,' added the March Hare, `that "I like what I get" is the same thing as "I get what I like"!' `You might just as well say,' added the Dormouse, who seemed to be talking in his sleep, `that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!' -Quotes from Alice in Wonderland Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

PRECISION of THOUGHT and LANGUAGE is essential to achieve the mathematical certainty that is needed if you are to have complete condence in your solutions to mathematical problems.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Method of Direct Proof

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Method of Direct Proof 1

Express the statement to be proved in the form

∀x ∈ D ,

if

P (x )

then

Q (x ).

(This step is often done mentally.)

2

Start the proof by supposing element of

D

x

(This step is often abbreviated

3

is a particular but arbitrarily chosen

P (x ) is true. Suppose x ∈ D and P (x ).)

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for which the hypothesis

Show that the conclusion

Q (x )

is true by using denitions,

previously established results, and the rules for logical inference.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Writing Proofs of Universal Statements

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Writing Proofs of Universal Statements (1) Copy the statement of the theorem to be proved on your paper. (2) Clearly mark the beginning of your proof with the word

Proof. (3) Make your proof self-contained.

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This means that you should explain the meaning of each variable used in your proof in the body of the proof. Thus you will begin proofs by introducing the initial variables and stating what kind of objects they are. At a later point in your proof, you may introduce a new variable to represent a quantity that is known at that point to exist. (4) Write your proof in complete, gramatically correct sentences (which may include symbols and abbreviations). updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Writing Proofs of Universal Statements (5) Keep your reader informed about the status of each statement in your proof.

If something is assumed, preface it with a word like Suppose or Assume. If it is still to be shown, preface it with words like, We must show that or In other words, we must show that.

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(6) Give a reason for each assertion in your proof.

Indicate the reason for each step of your proof using phrases such as by hypothesis, by denition of . . . , and by theorem . . . . (7) Include the little words and phrases that make the logic of your arguments clear.

Start the sentence by stating the reason why it follows or by writing Then, or Thus, or So, or Hence, or Therefore, or Consequently, or It follows that, and include the reason at the end of the sentence. If a sentence expresses a new thought or fact that does not follow as an immediate consequence of the preceding statement but is needed for a later part of a proof, introduce it by writing Observe that, or Note that, or But, or Now.

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(8) Display equations and inequalities. Discrete Mathematics for Computer Science

CS 30

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Direct Proofs Indirect Proofs

Methods of Proof

Try this ... Prove: If

k

is any odd integer and

m

is any even integer, then

k 2 + m2

is

odd.

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Discrete Mathematics for Computer Science

CS 30

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Direct Proofs Indirect Proofs

Methods of Proof

Try this ... Prove: If

k

is any odd integer and

m

is any even integer, then

k 2 + m2

is

odd.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Prove: If

n

is any even integer, then

(âˆ’1)n = 1.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Prove: If

n

is any even integer, then

(âˆ’1)n = 1.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Prove: The product of any two odd integers is odd.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Prove: The product of any two odd integers is odd.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Methods of Proof

Direct Proofs Indirect Proofs

Try this ... Use the quotient-remainder theorem with

d =3

to prove that the

product of any three consecutive integers is divisible by 3.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Proving Existential Statements

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Proving Existential Statements Examples: 1

Show that there is a positive integer that can be written as the sum of cubes of positive integers in two dierent ways.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Proving Existential Statements Examples: 1

Show that there is a positive integer that can be written as the sum of cubes of positive integers in two dierent ways. (A constructive existence proof )

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Proving Existential Statements Examples: 1

Show that there is a positive integer that can be written as the sum of cubes of positive integers in two dierent ways. (A constructive existence proof )

2

Show that there exist irrational numbers

x

and

y

such that

xy

is

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rational.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Proving Existential Statements Examples: 1

Show that there is a positive integer that can be written as the sum of cubes of positive integers in two dierent ways. (A constructive existence proof )

2

Show that there exist irrational numbers

x

and

y

such that

xy

is

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rational. (A nonconstructive existence proof )

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Disproving Universal Statements by Counterexamples The quotient of any two rational numbers is a rational number.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Disproving Universal Statements by Counterexamples The quotient of any two rational numbers is a rational number.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Disproving an Existential Statement Proving that the given statement is false is equivalent to proving its negation is true. Note: The negation of an existential statement is a universal statement.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Method of Proof by Contradiction 1

Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true.

2

Show that this supposition leads logically to a contradiction.

3

Conclude that the statement to be proved is true.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

An Example The square root of any irrational number is irrational.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

An Example The square root of any irrational number is irrational.

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

Method of Proof by Contraposition 1

Express the statement to be proved in the form then

2

Q (x ).

Rewrite this statement in the contrapositive form false then

3

∀x ∈ D ,

if

P (x )

(This step may be done mentally.)

P (x )

∀x ∈ D ,

if

Q (x )

is

is false. (This step may also be done mentally.)

Prove the contrapositive by a direct proof.

x is a (particular such that Q (x ) is false. (b) Show that P (x ) is false. (a) Suppose

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but arbitrarily chosen) element of D

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

An Example If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

Direct Proofs Indirect Proofs

An Example If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Direct Proofs Indirect Proofs

Methods of Proof

Relation between Proof by Contradiction and Proof by Contraposition For all integers

n,

if

n2

is odd then

n

is odd.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Direct Proofs Indirect Proofs

Methods of Proof

Relation between Proof by Contradiction and Proof by Contraposition For all integers

n,

if

n2

is odd then

n

is odd.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Methods of Proof

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Questions? See you next meeting!

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Discrete Mathematics for Computer Science

CS 30

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