Module #4 - Functions 4/6/2014Michael P. Frank / Kees van Deemter1 Lecture adapted from CS3511, Discrete Methods Kees van Deemter Slides originally adapted.

Module #4 - Functions

04/10/23 Michael P. Frank / Kees van Deemter 1

Lecture adapted from CS3511,Discrete Methods

Kees van Deemter

Slides originally adapted from Michael P. Frank’s Slides originally adapted from Michael P. Frank’s Course Based on the TextCourse Based on the Text

Discrete Mathematics & Its ApplicationsDiscrete Mathematics & Its Applications (5(5thth Edition) Edition)

by Kenneth H. Rosenby Kenneth H. Rosen



Adaptation of Module #4:1. Functions

Rosen 5Rosen 5thth ed., §1.8 ed., §1.8~45 slides, ~1 lecture~45 slides, ~1 lecture



Functions

• From calculus, you know the concept of a real-From calculus, you know the concept of a real-valued function valued function ff, , which assigns to each number which assigns to each number xxRR one particular value one particular value yy==ff((xx), where ), where yyRR..

• ExampleExample: : ff defined by the rule defined by the rule f(x)=xf(x)=x22

• The notion of a function can be generalized to the The notion of a function can be generalized to the concept of assigning elements of concept of assigning elements of anyany set to set to elements of elements of anyany set. set.

• Functions are also called operators.Functions are also called operators.



Functions

• Functions were the key concept in lambda Functions were the key concept in lambda calculus and Haskell:calculus and Haskell:

x.x+2x.x+2 is the function that sends any appropriate is the function that sends any appropriate argument x to the corresponding value x+2argument x to the corresponding value x+2

• After the Haskell definition After the Haskell definition f x = x+2,f x = x+2,

f denotes that same function f denotes that same function x.x+2x.x+2• Let’s be more precise about functions, working Let’s be more precise about functions, working

towards a classification of functionstowards a classification of functions



Function: Formal Definition

• A A functionfunction f from (or “mapping”) A to Bf from (or “mapping”) A to B ((ff::AABB) is an assignment of ) is an assignment of exactly oneexactly one element element ff((xx))BB to each element to each element xxA.A.

• Generalizations:Generalizations:– Functions of Functions of nn arguments: arguments:

ff: (A: (A11 x A x A22... x A... x Ann)) B. B.

– A A partial partial (non-(non-totaltotal)) function function ff assigns assigns zero or zero or oneone elements of elements of BB to each element to each element xxAA..



Graphs of Functions

• We can represent a function We can represent a function ff::AABB as a set of ordered as a set of ordered pairs pairs f f =={({(aa,,ff((aa)) | )) | aaAA}}..

• This makes This makes ff a a relationrelation between A and B: between A and B:f is a subset of f is a subset of A x B. A x B. But functions are special:But functions are special:– for every for every aaAA, there is at least one pair , there is at least one pair ((aa,,bb)). Formally: . Formally:

aaAAbbB((a,b)B((a,b)f) f) – for every for every aaAA, there is at most one pair , there is at most one pair ((aa,,bb)). Formally: . Formally:

a,b,c((a,b)a,b,c((a,b)f f (a,c) (a,c)f f b bc) c) • A relation over numbers can be represent as a set of A relation over numbers can be represent as a set of

points on a plane. (A point is a pair points on a plane. (A point is a pair ((x,yx,y)).).)– A function is then a curve (set of points), A function is then a curve (set of points),

with only one with only one yy for each for each xx. .



• Functions can be represented graphically in Functions can be represented graphically in several ways:several ways:

• •

AB

a b

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f

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PlotBipartite GraphLike Venn diagrams

A B



Functions that you’ve seen before

• A A setset SS over universe over universe UU can be viewed as a can be viewed as a function from the elements of function from the elements of U U to …to …



Still More Functions

• A A setset SS over universe over universe UU can be viewed as a can be viewed as a function from the elements of function from the elements of U U to …to …

… … {{TT, , FF}, saying for each element of }, saying for each element of UU whether it is in whether it is in SS. (This is called the . (This is called the characteristic functioncharacteristic function of of SS))

Suppose U={0,1,2,3,4}. ThenSuppose U={0,1,2,3,4}. Then SS={1,3}={1,3}

SS(0)=(0)=SS(2)=(2)=SS(4)=(4)=FF, , SS(1)=(1)=SS(3)=(3)=TT..




• A A set operator,set operator, such as such as or or , can be , can be viewed as a function from … to …viewed as a function from … to …




• A A set operatorset operator such as such as or or can be can be viewed as a function …viewed as a function …

… from (ordered) pairs of sets, … from (ordered) pairs of sets, to sets. to sets. – Example: Example: (({1,3},{3,4})) = {3}(({1,3},{3,4})) = {3}



A new notation

• Sometimes we write Sometimes we write YYXX to denote the set to denote the set FF of of allall possible functions possible functions ff: : XXYY..

• Thus,Thus, f f YYXX is another way of saying is another way of saying that that ff: : XXYY..

• (This notation is especially appropriate, (This notation is especially appropriate, because for finite because for finite XX, , YY, we have , we have ||FF| = || = |YY||||XX||.. ))



Some Function Terminology

• If If ff::AABB, and , and ff((aa)=)=b b (where (where aaAA & & bbBB), ), then we say:then we say:– AA is the is the domaindomain of of ff. . – BB is the is the codomaincodomain of of ff..– bb is the is the imageimage of of a a under under ff..– aa is a is a pre-imagepre-image of of bb under under f.f.

• In general, In general, bb may have more than 1 pre-image. may have more than 1 pre-image.

– The The rangerange RRBB of of f f is is RR={={bb | | aa ff((aa)=)=bb } }..

We also saythe signatureof f is A→B.



Range versus Codomain

• The range of a function may not be its whole The range of a function may not be its whole codomain.codomain.

• The codomain is the set that the function is The codomain is the set that the function is declareddeclared to map all domain values into. to map all domain values into.

• The range is the The range is the particularparticular set of values in the set of values in the codomain that the function codomain that the function actuallyactually maps elements maps elements of the domain to.of the domain to.

• (The range is the smallest set that could be used as (The range is the smallest set that could be used as its codomain.)its codomain.)



Choosing the right (co)domain

Consider the function f = λx.100/xConsider the function f = λx.100/xIs f a (total) function from Int to R?Is f a (total) function from Int to R?• f is a partial function from Int to Rf is a partial function from Int to R• f is a (total) function from Int-{0} to Rf is a (total) function from Int-{0} to R

Consider g = λx.√xConsider g = λx.√xIs g a (total) function from R to R?Is g a (total) function from R to R?g is a total function from R+ to RxRg is a total function from R+ to RxRe.g. g(4)= (2,-2)e.g. g(4)= (2,-2)



Images of Sets under Functions

• Given Given ff::AABB, and , and SSAA,,

• The The imageimage of of SS under under ff is the set of all is the set of all images (under images (under ff) of the elements of ) of the elements of SS..

ff((SS) :) : { {ff((ss) | ) | ssSS}} : : { {bb | | ssSS: : ff((ss)=)=bb}}..

• The range of The range of ff equals equalsthe image (under the image (under ff) of ...) of ...



Images of Sets under Functions

• Given Given ff::AABB, and , and SSAA,,

• The The imageimage of of SS under under ff is the set of all is the set of all images (under images (under ff) of the elements of ) of the elements of SS..

ff((SS) :) : { {ff((ss) | ) | ssSS}} : : { {bb | | ssSS: : ff((ss)=)=bb}}..

• The range of The range of ff equals equalsthe image (under the image (under ff) of ) of ff’s domain’s domain..



One-to-One Functions

• A function is A function is one-to-one (1-1)one-to-one (1-1), or , or injectiveinjective, or , or an an injectioninjection, iff every element of its range has , iff every element of its range has onlyonly 1 1 pre-image. pre-image. – Formally: given Formally: given ff::AAB,B,

“f“f is injective” :is injective” : ( (xx,,yy: : xxy y ff((xx))ff((yy))).).

• In other words: only In other words: only one element of the domain is element of the domain is mapped mapped toto any given any given oneone element of the range. element of the range.– In this case, domain & range have same cardinality. In this case, domain & range have same cardinality.

What about codomain?What about codomain?



• Codomain may be larger.Codomain may be larger.



One-to-One Illustration

• Are these relations one-to-one functions?Are these relations one-to-one functions?

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•Not even a function!



Sufficient Conditions for 1-1ness

• For functions For functions ff over numbers, we say: over numbers, we say:– ff is is strictlystrictly increasingincreasing iff iff x>y x>y ff((xx))>f>f((yy)) for all for all

x,yx,y in domain; in domain;– ff is is strictlystrictly decreasingdecreasing iff iff x>y x>y ff((xx))<f<f((yy)) for for

all all x,yx,y in domain; in domain;

• If If ff is either strictly increasing or strictly is either strictly increasing or strictly decreasing, then decreasing, then ff must be one-to-one. must be one-to-one.– Does the converse hold?Does the converse hold?



Onto (Surjective) Functions

• A function A function ff::AABB is is ontoonto or or surjectivesurjective or or a a surjectionsurjection iff its range is equal to its iff its range is equal to its codomain (codomain (bbBB, , aaAA: : ff((aa)=)=bb).).

• ConsiderConsider “country of birth of” “country of birth of”: A: AB,B,where A=people, B=countries. where A=people, B=countries. Is this a function? Is this a function? Is it an injection? Is it an injection? Is it a surjection?Is it a surjection?




• A function A function ff::AABB is is ontoonto or or surjectivesurjective or or a a surjectionsurjection iff its range is equal to its iff its range is equal to its codomain codomain

• ConsiderConsider “country of birth of” “country of birth of”: A: AB,B,where A=people, B=countries. where A=people, B=countries. Is this a function? Is this a function? Yes (always 1 c.o.b.) Yes (always 1 c.o.b.) Is it an injection? Is it an injection? No (many have same c.o.b.)No (many have same c.o.b.) Is it a surjection? Is it a surjection? Probably yes ..Probably yes ..




• A function A function ff::AABB is is ontoonto or or surjectivesurjective or or a a surjectionsurjection iff its range is equal to its iff its range is equal to its codomaincodomain

• In predicate logic:In predicate logic:




• A function A function ff::AABB is is ontoonto or or surjectivesurjective or or a a surjectionsurjection iff its range is equal to its iff its range is equal to its codomain. codomain.

• In predicate logic:In predicate logic:

bbBBaaAA ff((aa)=)=bb




• A function A function ff::AABB is is ontoonto or or surjectivesurjective or or a a surjectionsurjection iff its range is equal to its iff its range is equal to its codomain (codomain (bbBBaaAA ff((aa)=)=bb).).

• E.g.E.g., for domain & codomain Z, the , for domain & codomain Z, the function function x.x+1 is injective and surjective.x.x+1 is injective and surjective.



Claim: if f:ZClaim: if f:ZZ and f =Z and f =x.x+1 thenx.x+1 thenf is 1-to-1 and also onto.f is 1-to-1 and also onto. ((ZZ is the set of is the set of allall integers) integers)

• Proof that Proof that f is ontof is onto: Consider any arbitrary : Consider any arbitrary element a of Z. We have f(a-1)=a, where a element a of Z. We have f(a-1)=a, where a Z. Z.

• Proof that Proof that f is 1-to-1f is 1-to-1: Suppose f(u)=f(w)=a. In : Suppose f(u)=f(w)=a. In other words, u+1=a and w+1=a. It follows that other words, u+1=a and w+1=a. It follows that u=a-1 and w=a-1, so u=w.u=a-1 and w=a-1, so u=w.



Illustration of Onto

• Are these functions Are these functions onto onto their depicted co-their depicted co-domains?domains?

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• Are these functions Are these functions ontoonto??

onto

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• Are these functions 1-1?Are these functions 1-1?

onto

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• Are these functions 1-1?Are these functions 1-1?

not 1-1onto

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Bijections

• A function is said to be A function is said to be a one-to-one a one-to-one correspondencecorrespondence, or , or a a bijectionbijection iff it is iff it is bothboth one-to-one one-to-one and and onto.onto.



Two terminologies for talking about functions

1.1. injectioninjection = one-to-one = one-to-one

2.2. surjectionsurjection = onto = onto

3.3. bijectionbijection = one-to-one correspondence = one-to-one correspondence

3 = 1&23 = 1&2



Bijections

• For bijections For bijections f:Af:ABB, there exists a function , there exists a function that is the that is the inverse inverse of of ff, , written written f f 11: : BBAA

• Intuitively, this is the function that undoes Intuitively, this is the function that undoes everything that everything that ff does does

• Formally, it’s the unique function such that Formally, it’s the unique function such that

......



Bijections

• For bijections For bijections f:Af:ABB, there exists an , there exists an inverse inverse of of ff, written , written f f 11: : BBAA

• Intuitively, this is the function that undoes Intuitively, this is the function that undoes everything that everything that ff does does

• Formally, it’s the unique function such that Formally, it’s the unique function such that

(the identity function on (the identity function on AA))

AIff 1



Bijections

• Example 1: Let f: Example 1: Let f: ZZZZ be defined as be defined as f(x)= f(x)= x+1x+1. . What is fWhat is f1 1 ??

• Example 2: Let gExample 2: Let g: : ZN be defined as be defined as g(x)= g(x)= |x||x|. What is g. What is g1 1 ??



Bijections

• Example 1: Let f: Example 1: Let f: ZZZZ be defined as be defined as f(x)=x+1. f(x)=x+1. What is fWhat is f1 1 ??

• ff11 is the function (let’s call it h) is the function (let’s call it h) h: h: ZZZZ defined as h(x)=x-1. defined as h(x)=x-1.

• Proof: Proof: Ifh

h(f(x)) = (x+1)-1 = x



Bijections

• Example 2: Let gExample 2: Let g: : ZN be defined as be defined as g(x)=|x|g(x)=|x|. What is g. What is g1 1 ??

• This was a trick question:This was a trick question: there is no such there is no such function, since g is not a bijection:function, since g is not a bijection:There is no There is no functionfunction h such that h such that h(|x|)=x and h(|x|)=h(|x|)=x and h(|x|)=x x

• (NB There is a (NB There is a relationrelation h h for which this is true.)for which this is true.)



The Diagonalisation Method• Georg Cantor (1873): Can we compare the sizes Georg Cantor (1873): Can we compare the sizes

of of infiniteinfinite sets? sets?• Example: card(Example: card(N N ) = card({0) = card({0**11**})?})?

– Both are infiniteBoth are infinite– But is one larger than the other?But is one larger than the other?

• Cantor’s idea:Cantor’s idea:– The size (cardinality) of a set should not depend on The size (cardinality) of a set should not depend on

the the identityidentity of its elements of its elements– Two finite sets Two finite sets AA and and BB have the have the same sizesame size if we can if we can

pair the elements of pair the elements of AA with elements of with elements of BB– Formally: there exists a Formally: there exists a bijection bijection between between AA and and BB



Correspondences (Cont’d)• Example: LetExample: Let– N N be the set of natural numbers {1, 2, 3, …} be the set of natural numbers {1, 2, 3, …}

– EE be the set of even natural numbers { be the set of even natural numbers {22, , 44, , 66, …}, …}

• Using Cantor’s definition of size, we can show that Using Cantor’s definition of size, we can show that N N and and EE have the have the same sizesame size::– Bijection (!): Bijection (!): ff ( (nn) = ) = 22nn

• Intuitively, Intuitively, EE is smaller than is smaller than NN, but, but– Pairing each element of Pairing each element of NN with its corresponding with its corresponding

element in element in EE is possible, is possible, – So we declare these two set to be the same sizeSo we declare these two set to be the same size

– This even thoughThis even though E E NN ( (E E is a real subset ofis a real subset of NN ) )

663344222211

ff ( (nn))nn

… …



Countable sets• A set X is finite if it has n elements, A set X is finite if it has n elements, for some n in for some n in NN..

• A set is A set is countablecountable if either if either – It is It is finitefinite or or– It has the It has the same sizesame size as as NN, the natural numbers, the natural numbers

• For example,For example,– NN is countable, and so are all its subsets: is countable, and so are all its subsets:

– EE is countable is countable– {0,1,2,3} is countable{0,1,2,3} is countable is countableis countable

• How about supersets of How about supersets of NN??



An Even Stranger Example…• Let Let QQ be the set of positive rational numbers be the set of positive rational numbers

QQ = { m/n | m,n = { m/n | m,n NN } }

• Just like Just like EE, the set , the set QQ has the same size as has the same size as N N !!– We show this giving a We show this giving a bijectionbijection from from QQ to to N N – QQ is thus is thus countablecountable

• One way is to enumerate (i.e., to list) X’s One way is to enumerate (i.e., to list) X’s elements. For example, for X=elements. For example, for X=Q Q ::

– Pair the first element of X with 1 from Pair the first element of X with 1 from N N

And so on, making sure every member of And so on, making sure every member of QQ

appears only appears only onceonce in the list in the list



An Even Stranger Example… (Cont’d)

• To build a list with the elements of To build a list with the elements of QQ– make inf. matrix with all positive rational numbersmake inf. matrix with all positive rational numbers– ii -th row contains all numbers with numerator -th row contains all numbers with numerator ii– jj -th column has all numbers with denominator -th column has all numbers with denominator jj– ii //jj is in is in ii -th row and -th row and jj -th column-th column 1/1 1/2 1/3 1/4 1/5

2/1 2/2 2/3 2/4 2/5

3/1 3/2 3/3 3/4 3/5

4/1 4/2 4/3 4/4 4/5

5/1 5/2 5/3 5/4 5/5

. . .




• Now we turn the previous matrix into a listNow we turn the previous matrix into a list

• A bad way: begin list with first rowA bad way: begin list with first row– Since rows are infinite, we will never get to 2Since rows are infinite, we will never get to 2ndnd row! row!




• Instead, we list the elements along diagonals:Instead, we list the elements along diagonals:

1/1 1/2 1/3 1/4 1/5

2/1 2/2 2/3 2/4 2/5

3/1 3/2 3/3 3/4 3/5

4/1 4/2 4/3 4/4 4/5

5/1 5/2 5/3 5/4 5/5

. . ..

. .

We should, however, eliminate repeated elements




• We list elements along diagonals We list elements along diagonals w/o repetitionsw/o repetitions::

1/1 1/2 1/3 1/4 1/5

2/1 2/2 2/3 2/4

3/1 3/2 3/3

4/1 4/2

5/1

. . ..

. .

1/1, 2/1, 1/2, 3/1, 1/3, …



Uncountable sets• Some sets have no correspondence with Some sets have no correspondence with NN• These sets are simply too big!These sets are simply too big!

– They are not countable: we say They are not countable: we say uncountableuncountable

• Theorem:Theorem:– The set of real numbers between 0 and 1The set of real numbers between 0 and 1

(e.g., 0.244, 0.3141592323....) is (e.g., 0.244, 0.3141592323....) is uncountableuncountableCall this set RCall this set R0,10,1

(Some sets are even larger. “Serious” set theory is (Some sets are even larger. “Serious” set theory is all about theorems that concern infinite sets. – all about theorems that concern infinite sets. – Most of this is irrelevant for this course.)Most of this is irrelevant for this course.)



Finally: diagonalisation

Theorem: | RTheorem: | R0,10,1 | > |N|. | > |N|. ProofProof strategy: strategy:

| R| R0,10,1 |>=|N|. Suppose | R |>=|N|. Suppose | R0,10,1 |=|N| and derive |=|N| and derive a contradiction: Each member of Ra contradiction: Each member of R0,10,1 can can be written as a zero followed by a dot and be written as a zero followed by a dot and a countable sequence of digits. Suppose a countable sequence of digits. Suppose there existed a complete enumeration of R, there existed a complete enumeration of R, (using (using whatever whatever order) <e1,e2,e3,...>.order) <e1,e2,e3,...>.



Cantor’s diagonalisation trick:

(starting from an arbitrary list)(starting from an arbitrary list)

e1. 0.e1. 0.00000000000000000000000....000000000000000000000....e2. 0.0e2. 0.01100000000000000000000....00000000000000000000....e3. 0.82e3. 0.82000000000000000000000....0000000000000000000....e4. 0.171e4. 0.17100000000000000000000....000000000000000000..........



Now construct a Real number n that’s not in the enumeration:

nn’s first digit (after the dot) = ’s first digit (after the dot) = [e1’s first digit] + 1[e1’s first digit] + 1

nn’s second digit = [e2’s second digit] + 1 ...’s second digit = [e2’s second digit] + 1 ...GeneralGeneral:: nn’s i-th difit = [e-i’s i-th digit] + 1’s i-th difit = [e-i’s i-th digit] + 1 i: i: nn differs from e-i in its i-th digit differs from e-i in its i-th digitContradiction: <e1,e2,e3,...> is not a Contradiction: <e1,e2,e3,...> is not a

(complete) enumeration after all. (complete) enumeration after all. QEDQED



Adaptation of Module #3:2. Sets and the Russell Paradox

Rosen 5Rosen 5thth ed., §1.6 ed., §1.6especially ex. 30 on p. 86especially ex. 30 on p. 86



Basic Set Notations

• Set enumeration {a, b, c} Set enumeration {a, b, c}

and set-builder {and set-builder {xx||PP((xx)}.)}. relation, and the empty set relation, and the empty set ..

• Set relations =, Set relations =, , , , , , , , , , etc., etc.

• Venn diagrams.Venn diagrams.

• Cardinality |Cardinality |SS| and infinite sets | and infinite sets NN, , ZZ, , RR..

• Power sets P(Power sets P(SS).).



Axiomatic set theory

• Various axioms, e.g., saying that the union of two sets is Various axioms, e.g., saying that the union of two sets is also a setalso a set

• One key axiom: One key axiom: Given a Predicate P, construct a set. The Given a Predicate P, construct a set. The set consists of all those elements x such that P(x) is true.set consists of all those elements x such that P(x) is true.

• But, the resulting theory turns out to be But, the resulting theory turns out to be logically logically inconsistentinconsistent! ! – This means, there exist set theory propositions This means, there exist set theory propositions pp such that you can such that you can

prove that both prove that both pp and and pp follow logically from the axioms of the follow logically from the axioms of the theory!theory!

The conjunction of the axioms is a contradiction!The conjunction of the axioms is a contradiction!– This theory is fundamentally uninteresting, because any possible This theory is fundamentally uninteresting, because any possible

statement in it can be (very trivially) “proved” by contradiction!statement in it can be (very trivially) “proved” by contradiction!



This version of Set Theory is inconsistent

Russell’s paradox:Russell’s paradox:

• Consider the set that corresponds with the Consider the set that corresponds with the predicate predicate x x x x ::

S S = {= {x x | | xxxx }. }.

• Now ask: is Now ask: is SSSS??



Russell’s paradox• LLet et S S = {= {x x | | xxxx }. Is }. Is SSSS??• If If SSSS, then S is one of those objects , then S is one of those objects xx for which for which

xxx. x. In other wordsIn other words, , SSSSWith Proof by Contradiction, we have SWith Proof by Contradiction, we have SSS

• If If SSSS, then S is not one of those objects , then S is not one of those objects xx for for which which xxx. x. In other wordsIn other words, , SSSSWith Proof by Contradiction, we have SWith Proof by Contradiction, we have SSS

• We conclude that both We conclude that both SSSS nor nor SSSS• Paradox!Paradox!



• To avoid inconsistency, set theory must To avoid inconsistency, set theory must somehow changesomehow change

Bertrand Russell1872-1970



One technique to avoid the problem:

• Given a Given a setset S S and a predicate P, construct a and a predicate P, construct a new set, consisting of those elements x of S new set, consisting of those elements x of S such that P(x) is true.such that P(x) is true.

• You’ve seen this technique in use: You’ve seen this technique in use: Haskell’s list comprehensions allow us to Haskell’s list comprehensions allow us to write write [x | x [x | x [1..] , even x], but not simply [1..] , even x], but not simply [x | even x], [x | even x],



Another technique to avoid the problem:

• Russell’s paradox arises from the fact that Russell’s paradox arises from the fact that we can write we can write xxxx (or(or x xxx, for that matter)., for that matter).Forbid such expressions using Forbid such expressions using typestypes..

• You’ve seen this technique in use: You’ve seen this technique in use: Haskell’s use of typing.Haskell’s use of typing.Applied to the present case: give Applied to the present case: give a type a type that forbids it from relating two things of that forbids it from relating two things of the same type.the same type.



Our focus: computability

• We shall not worry about “saving” set We shall not worry about “saving” set theory from paradoxes like Russell’stheory from paradoxes like Russell’s

• Instead, we shall use the Russell paradox in Instead, we shall use the Russell paradox in a different settinga different setting

• Before we do this, we need to talk about Before we do this, we need to talk about computability and Turing Machines.computability and Turing Machines.



Module #4 - Functions 4/6/2014Michael P. Frank / Kees van Deemter1 Lecture adapted from CS3511, Discrete Methods Kees van Deemter Slides originally adapted.

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