Mathematics for Computer Science


 Malcolm Cross
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1 Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: Course website:
2 Functions: some intuitions A function (or mapping or transformation) is one of the most basic of all mathematical concepts. Intuitively, a function is something that takes input and turns it into some kind of output. There is a heavy computational flavour to the intuition. Indeed, when we introduce Turing Machines next week, we will think of Turing Machines as simple devices for computing functions. In this lecture we are going to see how this concept is made made mathematical precise, and will then use certain kinds of functions (namely bijective functions) to compare the sizes of sets. Next week we will look at functions from a computational perspective.
3 Where we are going today We begin by introducing the notion of an ordered pair and of a relations. We then define functions to be certain kinds of relations, and define three special types of functions: injective functions, surjective functions, and bijective functions. We then define a fundamental concept: when two sets have the same number of elements or (to use the official terminology) when they are equinumerous. We then compare the sizes of a number of infinite sets, with some surprising results.
4 Ordered pairs In set theory, the notation x, y is used to indicate that x and y are are a pair of items with x coming first and y second. That is, this notation specifies both the items involved (x and y) and the order in which they occur. That is x, y is not the same as {x, y}. The order is not specified in {x, y}; after all {x, y} = {y, x}. For ordered pairs we must have that: x, y = u, v iff x = u and y = v. This is the characteristic property of ordered pairs.
5 Aside on ordered pairs As I remarked last week, in set theory everything is defined purely in terms of sets and indeed, the notation x, y is usually defined as follows: x, y = def {{x}, {x, y}} This is not an obvious definition; it was first discovered by Kazimierz Kuratowski in 1921, replacing a more complex definition given by Norbert Wiener in The crucial thing to notice about this definition is that it works as we would expect ordered pairs to work. That is, it is not difficult to show that under this definition we have that: x, y = u, v iff x = u and y = v, which is what we require of any successful definition.
6 Binary relations A binary relation is defined to be a set of ordered pairs. More precisely, given a set X and a set Y, we define X Y to be the set of all ordered pairs x, y such that x X and y Y. The set X Y is called the Cartesian product of X and Y. A binary relation is simply a subset of X Y. X is called the domain of the relation, and Y is called the range.
7 Binary Relation Examples I Let X = {1, 2, 3} and Y = {a, b}. Then R 1 = { 1, a, 2, a } is a binary relation on these sets. Another (bigger) binary relation on these sets is R 2 = { 1, b, 2, b, 2, a }. Another bigger (and fact the biggest possible) binary relation on these sets is simply X Y itself. When dealing with small relations like these, it is common (and useful) to draw them using dots (to represent the entities) and arrows (to indicate which entities are related).
8 Binary Relation Examples II When defining binary relations it is very common for the domain and range of the relation to be the same set. That is, it is common to consider binary relations on X X (often written as X 2 ) for some set X. For example, let P be the set {Mia, Vincent, Marcellus}. Then L = { Vincent, Mia, Marcellus, Mia } is a binary relation (perhaps the loves relation) on X X. To give a more mathematical example, { x, y R R x y} is a binary relation on R R. It consists of all pairs of real numbers such the the first is less than or equal to the second. It can be represented on the Cartesian plane in the familiar way.
9 But what about functions? OK, but what about functions? At some stage in your career you have probably seem functions like f(x) = x 2, and are used to notation like f(2) = 4 and f(3) = 9. That is, you are probably used to being given definitions of functions f, g, h and so on, and used to the idea that such functions can be given arguments to produce results (or output). But what does this have to do with our discussion of relations? The answer is very simple...
10 Function A function with domain X and range Y is simply a special kind of binary relation. What makes it special? This: each x X is related to exactly one element y of Y. Suppose f is a function with domain X and range Y. Then we write f(x) to denote the unique element of Y that x is related to (that is, we simply sue the standard notation that you are used to from school maths). The image of a function f is {y Y : y = f(x) for some x X}.
11 Examples of functions (and nonfunctions) I Let X be {1, 2, 3, 4}, and let f be the following binary relation on X X: { 1, 2, 2, 3, 3, 2, 4, 3 }. It is easy to check that this binary relation is indeed a function. On the other hand suppose we let g be the following binary relation on X X: { 1, 2, 2, 3, 1, 3, 4, 3 }. This is clearly not a function. The binary relation { x, y R R y = x 2 } is a function. As you probably all saw at school, it looks like a parabola when drawn in the Cartesian plane. On the other hand, the binary relation { x, y R R y = x} is not a function (as positive real numbers have two square roots, one positive, the other negative).
12 Identity functions 1. A special (and very simple function) called the identity function can be defined on any set X. 2. The identity function id X, with domain X and range X is simply that function that maps each element of X to itself. 3. That is, for all x X, id X (x) = x.
13 Function composition The fundamental operation of functions is called function composition. In essence, this is the operation of plugging together two functions with compatible domains and ranges. More precisely, let A, B and C be sets, let g be a function with domain A and range B, and let f be a function with domain B and range C. Then fog, the composition of f with g is the function with domain A and range C defined by fog(a) = f(g(a)). That is, we first apply g to an elementa of A, and then apply f to the result, yielding an element of C.
14 Injective functions A function f is injective if and only if distinct elements of the domain map to distinct elements of the range. That is, a function is injective if and only if for all x, y in its domain such that x y, we have that f(x) f(y). Injective functions are often called onetoone functions.
15 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }, and let g be { a, b, b, c, c, b }. Then both f and g are functions, but f is injective and g is not. The function with domain and range R defined by y = x + 1 is injective. The function with domain and range R defined by y = x 2 is not injective.
16 Surjective functions A function f is said to be surjective if its image is its range. To put it another way, a surjective function is one where every element of the range has some element of the domain that is mapped to it. Surjective functions are often called onto functions.
17 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }, and let g be { a, b, b, c, c, b }. Then both f and g are functions, but f is surjective and g is not. The function with domain and range R defined by y = x 3 is surjective. The function with domain and range R defined by y = x 2 is not surjective.
18 Bijective functions A function is bijective if and only if it is both injective and surjective. That is, a function is a bijection if and only if it is both one to one and onto. Bijections have the following property: they can be turned around. More precisely, if f is a bijection with domain A and range B, then there is a bijection (called f 1 ) with domain B and range A such that f 1 of = id A. What does this mean and why is it it true?
19 Examples Let A be {a, b, c}, let f be { a, b, b, c, c, a }. As we have already seen, f is both injective and surjective. That is, f is bijective. What is f 1? The function with domain and range R defined by g(x) = x + 1 is bijective. What is g 1? For any set X, the identity function id X is a bijection from X to X.
20 Bijections and having the same size We are now ready to turn to the link between bijections and counting, and more generally, what it means for two sets to have the same size. Intuitively, when we count objects, what we are doing is establishing a bijection between a set of numbers and a set of objects. For example, if there are three doughnuts on the table, and I count them, what I have done is establish a bijection between the three doughnuts and the set {1, 2, 3}. More generally, if I show that there is a bijection between two sets, I have (even without counting) shown that they have the same size. This prompts the following definition...
21 Equinumerous sets Two sets are said to be equinumerous (that is, to have the same number of elements) if and only if there is a bijection between them. To put it another way, two sets are said to have the same size (the same cardinality) if and only if it is possible to map one to the other via a bijection. Sometimes we say that two sets can be put into one to one correspondence instead of saying that they are equinumerous. That is, this expression is just another common way of saying that there is a bijection between the two sets.
22 Equinumerous sets For example, {a, b, c} and {x, y, z} have the same number of elements. Why? Because it is easy to show that there is a bijection between these two sets. The elements can be paired off. Of course, we could say that these two sets have the same size by counting them, and saying that they both have three elements. But we don t need to count. The idea of having the same number of elements does not depend on the notion of number.
23 Comparing infinite sets Now that we know what having the same size means, we can start comparing sets. Now, this is a little boring for finite sets (we all learned to count finite sets when we were young). But matters are more interesting when we compare infinite sets. The most fundamental infinite set in set theory is N, the set of natural numbers. This is the set that contains all our usual counting numbers (0, 1, 2, 3, 4,... ) and nothing else. To get the ball rolling, let us call a set X countably infinite if it can be put into one to one correspondence with N (that is, if it is equinumerous with N).
24 The set of even numbers is countably infinite The set E of even numbers, that is, {0, 2, 4, 6, 8,...} is countably infinite. That is, there is a bijection with domain E and range N. The function is the obvious one: simply divide each even number by two. That is: we use the function f such that f(e) = e/2. It is easy to check that f is both injective and surjective. Thus while it is true that E N, it is also true that E and N have the same number of elements. E is countably infinite.
25 The set of integers is countably infinite The set of integers Z (that is, {... 3, 2, 1, 0, 1, 2, 3...} is also countably infinite. That is, there is a bijection with domain Z and range N. Here are the first few values of a suitable bijection f: f(0) = 0, f(1) = 1, f( 1) = 2, f(2) = 3, f( 2) = 4, f(3) = 5, f( 3) = 6, f(4) = 7, f( 4) = 8, Draw a diagram to see the pattern. That is, we have the funtion defined by f(z) = 2z, for all z 0, and f(z) = 2z 1 for z > 0. Thus while it is true that N Z, it is also true that Z and N have the same number of elements. Z is countably infinite.
26 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 0. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
27 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 0. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
28 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 1. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
29 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 2. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
30 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 3. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
31 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 4. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
32 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 5. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
33 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 6. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
34 N N is countably infinite 0, 3 1, 3 2, 3 3, , 2 1, 2 2, 2 3, , 1 1, 1 2, 1 3, , 0 1, 0 2, 0 3, 0... Let s count these ordered pairs. This one maps to 6. It is not particularly hard to write down an expression that captures this idea: j(m, n) = [(m + n) 2 + 3m + n]/2 And it is not particularly difficult to show that j is a bijection from N N to N.
35 Lots of interesting sets are countably infinite In particular, the set of rational numbers Q is countably infinite. This can be seen by an argument rather like the one we just saw for N N. In fact this might lead us to wonder are all infinite sets countably infinite? The answer to this is no. There are many bigger infinite sets. In fact, hre infinitely many sizes of ever everbigger infinities! Let s take a closer look...
36 Bigger infinities We are going to prove two main results The set of real numbers R (an infinite set( is not countably infinite it is bigger. For any set X, the set P(X) is bigger. We saw last week that this result holds for finite sets. This week we will see that it holds for arbitrary sets. A consequence of this result is if we start with an infinite set (say N) and then form its power set, we get a bigger infinite set. If we take the power set of that, we get a still bigger infinite set. And so am and so on and so on... the infinities keep getting bigger. The countably infinite is merely the bottom rung of the ladder of infinity.
37 The Diagonal argument Both arguments are proved using what have become known as diagonal arguments. This celebrated technique, was originally due to Georg Cantor, the father of set theory, who used it to prove the two results we shall now prove. As you will see, the two arguments have a slightly different form. However the underlying idea is the same. Nowadays the diagonal argument is used in many different forms in many different parts fo mathematics and computer science.
38 The real numbers R Recall from that every real number r can be represented by an infinite decimal number. For example: Can all these numbers be counted? That is, can we define a bijection form N to these numbers. The diagonal argument gives a systematic way of showing that no such function exists. If someone proposes such a function, we have a systematic method for showing it does not work.
39 The heart of the argument Suppose we have a function f with domain N that does count the real numbers. For example, suppose we have that: f(0) = f(1) = f(2) = f(3) = Define the following number. Before the decimal point it is 0. After the decimal point, its n + 1th decimal place is always 3, unless the n+1th place of f(n) is 3, in which case the is is 4. In this case, the number we build is: This number is not in the image of f, so f cannot be surjective, so it not a bijection.
40 Remark And that s the heart of diagonalisation. That is, someone proposes a suitable function that allegedly counts the set we are interested. We construct something (here a real number) that is not in the image of the function, thereby showing that the function cannot be surjective, and hence not bijective. We do so by working down the diagonal, systematically changing things. The technique takes it names from this idea.
41 Power sets are big! We are now going to show that any set X is always smaller than it power set P(X). The key word to observe here is the any ; we make no assumption that X is finite. We again show this by means of a (slightly different form) of the diagonal argument.
42 Some preliminary observations... First, recall that the result is true for finite sets. We proved last week that if a finite set X has n elements, then P(X) has 2 n elements. Second, it straightforward to see that for any X at all, whether X is finite or infinite, there is a bijection from X to a subset P(X). We simply use the function f that maps every y X to {y}. This means that P(X) can t possibly be smaller than X there is a clear sense in which X is embedded inside it. Still, none of this shows that there it is impossible (at least for infinite sets) to find a cunning way of bijectively mapping X to P(X). To prove this is impossible we need to be a bit smarter.
43 What we will do Let X be any set, and let g be a function from X to P(X). Again we will use diagonalisation to define a subset D g of X, called the diagonal set of g, that is is not in the image of g. That is we will show that g is not surjective, It s not obvious that we can do this after all, we know very little about X and g. But we can, as follows...
44 Diagonalisation Given any such function we define D g, the diagonal of g as follows: D g = {z X : z g(z)} Clearly D g X. That is, D g belongs to P(X). But for each z X: z D g iff z g(z). Hence D g g(z) for any z. Think about this carefully! And look at today s worksheet!
45 Consequences This argument shows that no function g from X to P(X) can be surjective, and hence no such function g can be a bijection. That is, for any set X we have that P(X) is bigger than X. In particular, P(N) is bigger than N. P(N) is certainly not countably infinite it is bigger. And P(P(N)) is bigger than P(N). And P(P(P(N))) is bigger than P(P(N)). And so on and so on: the infinities just keep on getting bigger and bigger and the process never stops...
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