Season 2 • Episode 02 • Proof or spoof ?

Season 2 • Episode 02 • Proof or spoof ?

⁰

Proof or spoof ?

Season 2

Episode 02 Time frame 1 period

Objectives :

•

^{Workon the onept of proof and disoversome methods.}

Materials :

•

^{Titles and} explanations for some methods of proof.

1 – Matching game 25 mins

Studentswork ingroups of4or5.Theyare given the titlesand explanationsorexamples

for dierent kinds of proof, all mixed up. They have toput the title bak with the right

explanation or example.

2 – Find the valid ones 30 mins

Eah group has tond out whih methods are valid and whih ones are not.

Proof or spoof ?

Season 2

Episode 02

Document Explanations

Type of proof

^{Valid Not valid}

AnoshootofProof byIndution, onemayassumethe resultistrue.Therefore itistrue.

Type of proof

^{Valid Not valid}

Althoughnot a formalproof, avisualdemonstrationofa mathematialtheorem is some-

times alled a proof without words. It isoften used toprove the Pythagorean theorem.

Type of proof

^{Valid Not valid}

Alsoalledproofbyexample,itistheexhibitionofaonreteexamplewithapropertyto

show that something having that property exists. Joseph Liouville, for instane, proved

the existeneof transendental numbers by onstrutingan expliit example.

Type of proof

^{Valid Not valid}

Multiplyboth expressions by zero, e.g.,

1 = 2 1 × 0 = 2 × 0

0 = 0

Sine the nal statement is true, sois the rst.

Type of proof

^{Valid Not valid}

In this type of proof, the onlusion is established by logially ombining the axioms,

denitions, and earlier theorems. For example, it an be used to establish that the sum

of twoeven integersis always even :

Consider two even integers

x

^and

y

^{. Sine they are even, they an be written}

as

x = 2a

^and

y = 2b

respetively for integers

a

^and

b

^{.Then the sum}

x + y = 2 a + 2 b = 2( a + b )

^{. From this it islear}

x + y

^has

2

^{asa fator and therefore}

is even, so the sum of any two even integers is even. This proof uses only

denition of even integers and the distributionlaw.

Type of proof

^{Valid Not valid}

AN ARGUMENTMADE INCAPITAL LETTERSISCORRECT. THEREFORE, SIM-

PLY RESTATE THE PROPOSITION YOU ARE TRYING TO PROVE IN CAPITAL

LETTERS, ANDIT WILL BECORRECT!!!!!1 (USETYPOSAND EXCLAMATION

Season 2 • Episode 02 • Proof or spoof ?

²

Type of proof

^{Valid Not valid}

Bywritingwhat seemstobeanextensive proofand thensmearinghoolate tostainthe

most ruialparts, thereader willassume thatthe proof isorretsoas nottoappear to

bea fool.

Type of proof

^{Valid Not valid}

In this importanttype of proof rst a base ase is proved, and then anindutiverule

is used to prove a (often innite) series of other ases. Sine the base ase is true, the

innity of other ases must also be true, even if all of them annot be proved diretly

beauseof their innite number.

Its priniplestatesthat : LetN

= { 1, 2, 3, 4, . . . }

^{be theset ofnaturalnumbers and}

P (n)

bea mathematialstatement involving the naturalnumber

n

^{belongingto Nsuh that}

(i)

P (1)

^{istrue, i.e.,}

P ( n )

^{is true for}

n = 1

^;

(ii)

P (n + 1)

^{istruewhenever}

P (n)

^istrue,i.e.,

P (n)

^{istrue impliesthat}

P (n + 1)

^istrue.

Then

P (n)

^{istrue forall naturalnumbers}

n

^.

Type of proof

^{Valid Not valid}

Remember, something isnot true when itsproof has been veried,it istrue as longas it

has not been disproved. Forthis reason,the best strategy is tolimitas muh aspossible

the numberof people with the needed ompeteneto understand your proof.

Besuretoinludeveryomplexelementsinyourproof.Innitenumbersofdimensions,hy-

peromplexnumbers,indeterminateforms,graphs,referenestoveryoldbooks/movies/bands

that almost nobody knows, quantum physis, modallogi, and hess opening theory are

to be inluded in the thesis. Make sentenes in Latin, Anient Greek, Sanskrit, Ithkuil,

and invent languages.

Again, the goal :nobodymust understand, and this way, nobody an disprove you.

Type of proof

^{Valid Not valid}

Inthistypeofproof,(alsoknownasredutioadabsurdum,Latinforbyredutiontoward

the absurd), it is shown that if some statement were so, a logial ontradition would

our, hene the statement must be not so. This methodis one of the most prevalent of

mathematial proofs. A famous example shows that

√ 2

^{is anirrationalnumber:}

Suppose that

√ 2

^{is a rational number, so}

√

2 = ^a b

^where

a

^and

b

^{are non-}

zero integers with noommon fator (denition of a rational number). Thus,

b √

2 = a

^{.Squaringbothsidesyields}

2b ² = a ²

^{.Sine2dividesthelefthandside,}

2 must also divide the right hand side (asthey are equal and both integers).

So

a ²

^{iseven, whihimpliesthat}

a

^{must alsobeeven.Sowe anwrite}

a = 2 c

^,

where

c

^{is alsoaninteger.}Substitutionintothe originalequationyields

2b ² = (2c) ² = 4c ²

^{. Dividing both sides by 2 yields}

b ² = 2c ²

^{. But then, by the same}

argument as before, 2 divides

b ²

^{, so b must be even. However, if}

a

^and

b

^are

both even, they sharea fator, namely2.This ontraditsour assumption,so

we are fored to onlude that

√ 2

^{isan irrationalnumber.}

Season 2 • Episode 02 • Proof or spoof ?

³

Type of proof

^{Valid Not valid}

"I believeassertion A tohold, therefore itdoes. Q.E.D."

Type of proof

^{Valid Not valid}

Ifenoughpeoplebelievesomethingtobetrue,thenitmustbeso.Foreven moreemphati

proof, one an use the similar Proof by a BroadConsensus.

Either kind of proof an be ombined with other types of proof (suh as Proof by Re-

petition and Proof by Intimidation; e.g., "A Broad Consensus of Sientists ...") when

required.

Type of proof

^{Valid Not valid}

It establishes the onlusion if

p

^then

q

^{by proving the equivalent} ontrapositive state- ment if not

q

^{then not}

p

^.

Type of proof

^{Valid Not valid}

The propositionis true due tothe lak of a ounterexample. For whenyouknowyouare

right and that you don't give a shit about what others may thinkof you.

Type of proof

^{Valid Not valid}

In this type of proof, the onlusion is established by dividing it intoa nite number of

ases and proving eah one separately. The numberof ases sometimes an beome very

large.Forexample,therstproofofthefourolortheoremwasaproofbyexhaustionwith

1,936 ases. This proof was ontroversial beause the majority of the ases were heked

byaomputerprogram,notbyhand.Theshortestknownproofofthefourolourtheorem

today stillhas over 600 ases.

Type of proof

^{Valid Not valid}

Makeiteasieronyourself by leavingitup tothereader.Afterall,ifyouangureitout,

surely they an. Examples :

•

^{"The readermay easilysupply the} details."

•

^{"The other 253 ases are} analogous."

•

^{"The proof isleft as anexerise for the} reader."

•

^{"The proof isleft as anexerise for the} marker." (Guaranteed to work inan exam.)

Type of proof

^{Valid Not valid}

SineAugustissuhagoodtimeofyear,then no-onewilldisagreewithaproofpublished

then,and thereforeitistrue.Ofoursethe onverse isalsotrue, i.e.Januaryisshite,and

all the logi inthe world willnot prove your statement.

Season 2 • Episode 02 • Proof or spoof ?

⁴

Type of proof

^{Valid Not valid}

Reduingproblemstodiagramswithlotsofarrows.Thisisrelatedtoproofbyomplexity.

Type of proof

^{Valid Not valid}

Ifthere isaonsensusonatopi.and youdisagree, thenyouarerightbeausepeopleare

stupid. See global warming septis, reationist, tobao ompanies, et., for appliation

of this proof.

Type of proof

^{Valid Not valid}

Besuretoprovidesomedistrationwhileyougoonwithyourproof,e.g.,somethird-party

announes, a re alarm (a fake one would do, too)or the end of the universe. You ould

alsoexlaim, "Look!A distration!",meanwhilepointingtowardsthe nearest brik wall.

Be sure towipethe blakboardbeforethe distrationispresumably oversoyouhavethe

whole board for your nal onlusion.

Don't be intimidated if the distration takes longer than planned and simply head over

to the next proof.

An example isgiven below.

1

^{. Look behindyou!}

2

^{. ... and proves the existene of ananswer for2 +2.}

3

^{. Look! A} three-headed monkey over there!

4

^{. ... leaves 5 asthe only result of 2 +2.}

5

^{. Therefore 2+ 2= 5. Q.E.D.}

Type of proof

^{Valid Not valid}

Suppose

P ( n )

^{is astatement.}

1

^{. Prove true for}

P (1)

^.

2

^{. Prove true for}

P (2)

^.

3

^{. Prove true for}

P (3)

^.

4

^{. Therefore}

P (n)

^{is true for all}

n

^.

Type of proof

^{Valid Not valid}

If Jak Bauer says something is true, then it is. No ifs, ands, or buts about it. End of

disussion.

Season 2 • Episode 02 • Proof or spoof ?

⁵

Type of proof

^{Valid Not valid}

If you say something is true enough times, then it is true. If you say something is true

enough times, then it is true. If you say something is true enough times, then it is true.

If you say something is true enough times, then it is true. If you say something is true

enough times, thenit is true. Ifyousay something istrue enough times, then itis true.

Exatly how many times one needs to repeat the statement for it to be true is debated

widely inaademi irles.Generally,thepointisreahed whenthosearounddiethrough

boredom.

eg. let

A = B

^sine

A = B

^and

A = B

^and

A = B

^and

A = B

^and

A = B

^and

A = B

and

A = B

^and

A = B

^and

A = B

^and

A = B

^then

A = B

^.

Type of proof

^{Valid Not valid}

If you prove your laim for one ase, and make sure to restrit yourself to this one, you

thus avoid any ase that ould ompromise you. You an hope that people won't notie

the omission.

Example : Prove the four-olor theorem. Take a map of only one region.Only 1 olor is

needed toolor it and 1 isless than4. End of the proof.

Ifsomeonequestionsthe ompletenessofthe proof,othersmethodsofproofsan beused.

Proof or spoof ?

Season 2

Episode 02

Document Types of proof

•

^{Visual proof}

•

^{Proof by} Multipliative Identity

•

^{Proof by Restrition}

•

^{Proof by Assumption}

•

^{Proof by Belief}

•

^{Proof by} Constrution

•

^{Proof by Cases}

•

^{Proof by Complexity}

•

^{Proof by (a Broad) Consensus}

•

^{Proof by August}

•

^{Proof by Default}

•

^{Proof by T}ransposition

•

^{Proof by Delegation}

•

^{Diret proof}

•

^{Proof by Choolate}

•

^{Proof by Dissent}

•

^{Proof by Distration}

•

^{Proof by Exhaustion}

•

^{Proof by Engineer's Indution}

•

^{Proof by Diagram}

•

^{Proof by} Mathematial Indution

•

^{Proof by Jak Bauer}

•

^{Proof by Repetition}

•

^{Proof by} ontradition

Season 2 • Episode 02 • Proof or spoof ?

⁷

Document 1

^{Methodsof proof : titles and} explanations

Direct proof

In this type of proof, the onlusion is established by logially ombining the axioms,

denitions, and earlier theorems. For example, it an be used to establish that the sum

of twoeven integersis always even :

Consider two even integers

x

^and

y

^{. Sine they are even, they an be written}

as

x = 2 a

^and

y = 2 b

respetively for integers

a

^and

b

^{.Then the sum}

x + y = 2a + 2b = 2(a + b)

^{. From this it islear}

x + y

^has

2

^{asa fator and therefore}

is even, so the sum of any two even integers is even. This proof uses only

denition of even integers and the distributionlaw.

Proof by mathematical induction

In this importanttype of proof rst a base ase is proved, and then anindutiverule

is used to prove a (often innite) series of other ases. Sine the base ase is true, the

innity of other ases must also be true, even if all of them annot be proved diretly

beauseof their innite number.

Its priniplestatesthat : LetN

= { 1, 2, 3, 4, . . . }

^{be theset ofnaturalnumbers and}

P (n)

bea mathematialstatement involving the naturalnumber

n

^{belongingto Nsuh that}

(i)

P (1)

^{istrue, i.e.,}

P (n)

^{is true for}

n = 1

^;

(ii)

P (n + 1)

^{istruewhenever}

P (n)

^istrue,i.e.,

P (n)

^{istrue impliesthat}

P (n + 1)

^istrue.

Then

P ( n )

^{istrue forall naturalnumbers}

n

^.

Proof by transposition

It establishes the onlusion if

p

^then

q

^{by proving the equivalent} ontrapositive state- ment if not

q

^{then not}

p

^.

Proof by contradiction

Inthistypeofproof,(alsoknownasredutioadabsurdum,Latinforbyredutiontoward

the absurd), it is shown that if some statement were so, a logial ontradition would

our, hene the statement must be not so. This methodis one of the most prevalent of

mathematial proofs. A famous example shows that

√ 2

^{is anirrationalnumber:}

Suppose that

√ 2

^{is a rational number, so}

√

2 = ^a _b

^where

a

^and

b

^{are non-}

zero integers with noommon fator (denition of a rational number). Thus,

b √

2 = a

^{.Squaringbothsidesyields}

2 b ² = a ²

^{.Sine2dividesthelefthandside,}

2 must also divide the right hand side (asthey are equal and both integers).

So

a ²

^{iseven, whihimpliesthat}

a

^{must alsobeeven.Sowe anwrite}

a = 2c

^,

where

c

^{is alsoaninteger.}Substitutionintothe originalequationyields

2 b ² = (2 c ) ² = 4 c ²

^{. Dividing both sides by 2 yields}

b ² = 2 c ²

^{. But then, by the same}

argument as before, 2 divides

b ²

^{, so b must be even. However, if}

a

^and

b

^are

both even, they sharea fator, namely2.This ontraditsour assumption,so

we are fored to onlude that

√ 2

^{isan irrationalnumber.}

Season 2 • Episode 02 • Proof or spoof ?

⁸

Proof by construction

Alsoalledproofbyexample,itistheexhibitionofaonreteexamplewithapropertyto

show that something having that property exists. Joseph Liouville, for instane, proved

the existeneof transendental numbers by onstrutingan expliit example.

Proof by exhaustion

In this type of proof, the onlusion is established by dividing it intoa nite number of

ases and proving eah one separately. The numberof ases sometimes an beome very

large.Forexample,therstproofofthefourolortheoremwasaproofbyexhaustionwith

1,936 ases. This proof was ontroversial beause the majority of the ases were heked

byaomputerprogram,notbyhand.Theshortestknownproofofthefourolourtheorem

today stillhas over 600 ases.

Visual proof

Althoughnot a formalproof, avisualdemonstrationofa mathematialtheorem is some-

times alled a proof without words. It isoften used toprove the Pythagorean theorem.

Proof by Multiplicative Identity

Multiplyboth expressions by zero, e.g., let's prove that 1

1 = 2

^:

1 = 2

1 × 0 = 2 × 0 0 = 0

Sine the nal statement is true, sois the rst.

Proof by August

SineAugustissuhagoodtimeofyear,then no-onewilldisagreewithaproofpublished

then,and thereforeitistrue.Ofoursethe onverse isalsotrue, i.e.Januaryisshite,and

all the logi inthe world willnot prove your statement.

Proof by Assumption

AnoshootofProof byIndution, onemayassumethe resultistrue.Therefore itistrue.

Proof by Belief

"I believeassertion A tohold, therefore itdoes. Q.E.D."

Proof by Cases

AN ARGUMENTMADE INCAPITAL LETTERSISCORRECT. THEREFORE, SIM-

PLY RESTATE THE PROPOSITION YOU ARE TRYING TO PROVE IN CAPITAL

LETTERS, ANDIT WILL BECORRECT!!!!!1 (USETYPOSAND EXCLAMATION

Season 2 • Episode 02 • Proof or spoof ?

⁹

Proof by Chocolate

Bywritingwhat seemstobeanextensive proofand thensmearinghoolate tostainthe

most ruialparts, thereader willassume thatthe proof isorretsoas nottoappear to

bea fool.

Proof by Complexity

Remember, something isnot true when itsproof has been veried,it istrue as longas it

has not been disproved. Forthis reason,the best strategy is tolimitas muh aspossible

the numberof people with the needed ompeteneto understand your proof.

Besuretoinludeveryomplexelementsinyourproof.Innitenumbersofdimensions,hy-

peromplexnumbers,indeterminateforms,graphs,referenestoveryoldbooks/movies/bands

that almost nobody knows, quantum physis, modallogi, and hess opening theory are

to be inluded in the thesis. Make sentenes in Latin, Anient Greek, Sanskrit, Ithkuil,

and invent languages.

Again, the goal :nobodymust understand, and this way, nobody an disprove you.

Proof by (a Broad) Consensus

Ifenoughpeoplebelievesomethingtobetrue,thenitmustbeso.Foreven moreemphati

proof, one an use the similar Proof by a BroadConsensus.

Either kind of proof an be ombined with other types of proof (suh as Proof by Re-

petition and Proof by Intimidation; e.g., "A Broad Consensus of Sientists ...") when

required.

Proof by Default

The propositionis true due tothe lak of a ounterexample. For whenyouknowyouare

right and that you don't give a shit about what others may thinkof you.

Proof by Delegation

Makeiteasieronyourself by leavingitup tothereader.Afterall,ifyouangureitout,

surely they an. Examples :

•

^{"The readermay easilysupply the} details."

•

^{"The other 253 ases are} analogous."

•

^{"The proof isleft as anexerise for the} reader."

•

^{"The proof isleft as anexerise for the} marker." (Guaranteed to work inan exam.)

Proof by Diagram

Reduingproblemstodiagramswithlotsofarrows.Thisisrelatedtoproofbyomplexity.

Proof by Dissent

Ifthere isaonsensusonatopi.and youdisagree, thenyouarerightbeausepeopleare

stupid. See global warming septis, reationist, tobao ompanies, et., for appliation

of this proof.

Season 2 • Episode 02 • Proof or spoof ?

¹⁰

Proof by Distraction

Besuretoprovidesomedistrationwhileyougoonwithyourproof,e.g.,somethird-party

announes, a re alarm (a fake one would do, too)or the end of the universe. You ould

alsoexlaim, "Look!A distration!",meanwhilepointingtowardsthe nearest brik wall.

Be sure towipethe blakboardbeforethe distrationispresumably oversoyouhavethe

whole board for your nal onlusion.

Don't be intimidated if the distration takes longer than planned and simply head over

to the next proof.

An example isgiven below.

1

^{. Look behindyou!}

2

^{. ... and proves the existene of ananswer for2 +2.}

3

^{. Look! A} three-headed monkey over there!

4

^{. ... leaves 5 asthe only result of 2 +2.}

5

^{. Therefore 2+ 2= 5. Q.E.D.}

Proof by Engineer’s Induction

Suppose

P (n)

^{is astatement.}

1

^{. Prove true for}

P (1)

^.

2

^{. Prove true for}

P (2)

^.

3

^{. Prove true for}

P (3)

^.

4

^{. Therefore}

P (n)

^{is true for all}

n

^.

Proof by Jack Bauer

If Jak Bauer says something is true, then it is. No ifs, ands, or buts about it. End of

disussion.

Proof by Repetition

If you say something is true enough times, then it is true. If you say something is true

enough times, then it is true. If you say something is true enough times, then it is true.

If you say something is true enough times, then it is true. If you say something is true

enough times, thenit is true. Ifyousay something istrue enough times, then itis true.

Exatly how many times one needs to repeat the statement for it to be true is debated

widely inaademi irles.Generally,thepointisreahed whenthosearounddiethrough

boredom.

eg. let

A = B

^sine

A = B

^and

A = B

^and

A = B

^and

A = B

^and

A = B

^and

A = B

and

A = B

^and

A = B

^and

A = B

^and

A = B

^then

A = B

^.

Proof by Restriction

If you prove your laim for one ase, and make sure to restrit yourself to this one, you

thus avoid any ase that ould ompromise you. You an hope that people won't notie

the omission.

Example : Prove the four-olor theorem. Take a map of only one region.Only 1 olor is

needed toolor it and 1 isless than4. End of the proof.