Notes for Science and Engineering Foundation Discrete Mathematics

Transcription

1 Notes for Science and Engineering Foundation Discrete Mathematics by Robin Whitty BSc PhD CMath FIMA 2009/2010

2 Contents 1 Introduction The Laws of Arithmetic Polynomial Arithmetic Laws of arithmetic for polynomials Powers of polynomials Matrix Arithmetic Motivation Addition and subtraction Scalar multiplication Vector Multiplication Matrix Multiplication Matrix Operations Matrices Addition Scalar multiplication Matrix multiplication Laws of arithmetic for matrices Matrix transpose Propositional Logic Arithmetic with propositions Disjunction: OR, (like + ) i

3 ii CONTENTS Conjunction: AND, (like ) Logical equivalence: (like = ) Negation: (like unary ) Laws of arithmetic for propositions Normal Form Truth Tables Enumerated Form Arithmetic Tables Logic Functions Proving Logical Equivalence Logical Implication Set Theory The Membership Predicate Specifying a set The equality and subset predicates A set of cardinality n has 2 n subsets The Arithmetic of sets Union, (like ) Intersection, (like ) Difference:\ (like ) Laws of arithmetic for propositions Venn Diagrams and Region Tables The Universal set Venn diagrams and complementation The complementation laws Venn diagrams for two sets Venn diagrams for three sets and truth tables again Region tables Cartesian Product and Relations 51

4 CONTENTS The Cartesian Product Laws of arithmetic and closure Relations Graphs of relations Properties of relations Special types of relations

5 2 CONTENTS

6 50 CONTENTS

7 Chapter 8 Cartesian Product and Relations Since Cartesian geometry gave us the idea of the x-y plane, over 400 years ago, the idea of working with pairs of numbers has become a central to much of mathematics. The pairs (x, y) that are coordinates of points in the plane consist of real numbers from the setrand are not of direct concern in discrete mathematics; but we will see that we can make pairs using any sets. 8.1 The Cartesian Product Suppose we have a universal setu and subsets A and B ofu. The cartesian product of A and B, denoted by A B and said A cross B, is defined to be the set of all pairs (a, b) with a A and b B: A B={(a, b) a A b B}. Note that the pairs are ordered: (a, b) (b, a) unless a=b. So, in particular, {a, b} is not the same as (a, b): one is a set of cardinality 2 and the order in which we list the elements does not matter; the other is a pair and the order does matter. E.g. if A={1, 2, 3} and B={p, q} then (a) A B={(1, p), (1, q), (2, p), (2, q), (3, p), (3, q)}; (b) B A={(p, 1), (p, 2), (p, 3), (q, 1), (q, 2), (q, 3)}; (c) A A={(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}; (d) B B={(p, p), (p, q), (q, p), (q, q)}; (e) (B B) B={ ( (p, p), p ), ( (p, p), q ), ( (p, q), p ), ( (p, q), q ), ( (q, p), p ), ( (q, p), q ), ( (q, q), p ), ( (q, q), q ) }; (f) B (B B)={ ( p, (p, p) ), ( q, (p, p) ), ( p, (p, q) ), ( q, (p, q) ), ( p, (q, p) ), ( q, (q, p) ), ( p, (q, q) ), ( q, (q, q) ) }; 51

8 52 CHAPTER 8. CARTESIAN PRODUCT AND RELATIONS We may visualise the cartesian product as a set of points in the plane, as in figure 8.1 for A B in the above example. We should be careful about interpreting this picture geometrically, however; for instance, the distance between the different points has no meaning. Later on we shall make a quite different visualisation in which pairs are represented by arrows instead of points. Figure 8.1: graphical representation of the set A B, A={1, 2, 3}, B={p, q}. The rectangle of points in figure 8.1 has size 2 3, so it contains 6 points in total. In general, if A contains m elements (recall that we write this as A =m) and B contains n elements, then A B contains m n elements: A B = A B. (There is a link to matrices here: a matrix with m rows and n columns has size m n (m by n) and contains m n entries.) Laws of arithmetic and closure In chapter 6 we met the set operations union, intersection cap and set difference \; then in chapter 7 we met the complement c operation. These operations all related to the arithmetic of propositions, although\, like disobeyed all the rules: associative, commutative and distributive. Now we have another set operation:. We shall see that it too disobeys all the rules. And in fact it has a worse failing, one which excludes from being considered as an arithmetic operation at all! We will check the laws of arithmetic for in a slightly different order than usual, looking back to the example above: Commutative laws: Does (A B) C=A (B C)? NO: we saw in example (a) that (1, p) A B but in (b) (1, p) B A; Associative law: Does (A B) C=A (B C)? NO, not even if A, B and C are all the same set: we saw in example (e)

9 8.2. RELATIONS 53 that ( (p, p), p ) (B B) B but in (f) we saw thatb (B B) contained ( p, (p, p) ) which is not the same; Distributive law: Does A (B C)=(A B) (A C)? NO: suppose A, B and C are all the set{a}. Then A (B C)={a} ( {a} {a} ) ={a} { (a, a) } =, but (A B) (A C)= ( {a} {a} ) ( {a} {a} ) ={a} { a}= { (a, a) }. In the last example, A (B C)= because A, B and C contain single elements from some universal setu but B C contains pairs of elements and these will not generally be elements ofu. Until now, all the operations we have met have stayed withinu : the sum of two polynomials is a polynomial; the sum of two matrices is a matrix; P Q is a proposition; A B is a subset ofu. But A B is not a subset ofu. We say that an operation is closed overu if its result always again belongs tou. For sets, we can think of this as saying that the operation keeps within the box of a Venn diagram;,,\and c all turn regions of the Venn diagram into new regions. But does not give a region of the Venn diagram. For this reason we do not think of it as an arithmetic operator at all (computer scientists sometimes call it a constructor ). E.g. (a)z Z={(x, y) x Z y Z} Z; (b) (Z Z) Z={ ( (x, y), z ) x, y, z Z} Z Z Z; (c)r R={(x, y) x, y R} R; The set of pairs of integersz Z is usually denotedz 2 (as though it was multiplication but it definitely is not multiplication!) Actually the sets (Z Z) Z andz (Z Z) are both usually denotedz 3 even though they are different sets! In applications, there is often no difference between ( (x, y), z ) and ( x, (y, z) ). This is especially true forr 3 which denotes three-dimensional space: it does not matter if we look at the x-y plane and then include the z axis, or look at the x axis and then include the y-z plane. Einstein taught us that this remains true forr 4 : time, the fourth dimension, is no different from length, breadth and height. 8.2 Relations If A is any set then we will write A 2 for the cartesian product A A. The subsets of A 2 are given a special name: they are called relations on A.

10 54 CHAPTER 8. CARTESIAN PRODUCT AND RELATIONS E.g. (a) The subset ofz 2 which pairs numbers with their squares: R 1 ={(x, y) y= x 2 } = { (0, 0), (1, 1), ( 1, 1), (2, 4), ( 2, 4) (3, 9), ( 3, 9),... }. (b) If A={a, b, c}, the subset of A 2 consisting of pairs in alphabetical order: R 2 ={(x, y) x comes before y in the alphabet} = { (a, b), (a, c), (b, c) }. (c) If A={a, b, c}, the subset of A 2 consisting of pairs not in alphabetical order: R 3 ={(x, y) x does not come before y in the alphabet} = { (a, a), (b, a), (b, b), (c, a), (c, b), (c, c) }. Note that R 2 R 3 = A 2. In other words, R 3 = R 2 c, as we would expect since the membership predicate for R 3 is the negation of the membership predicate for R 2. Some textbooks allow relations between two different sets. For example, if A is the set of living men and B is the set of living women then we could define R A B= { (x, y) x is legally married to y }. However, we will call this a mapping and deal with it later Graphs of relations We can represent a relation R A 2 diagrammatically: each element of A is represented by a point, called a vertex; each pair (a, b) R is represented by an arrow, called an edge, joining vertex a to vertex b. This representation is valid for any set A, even if it is infinite, likez. But we will only consider cases where A is a small finite set, say five or six elements at most. E.g. (a) The relation R 1 on A={a, b, c, d, e} given by R 1 = { (a, b), (b, a), (b, c), (b, d), (c, d), (c, e), (d, a), (d, e) } is shown top-left in figure 8.2. (b) The relation R 2 on A={a, b, c, d} given by R 2 = { (a, a), (b, a), (b, b), (b, c), (b, d), (c, a), (c, c), (c, d), (d, a), (d, d) }

11 8.2. RELATIONS 55 Figure 8.2: graphical representations of four relations. is shown top-right in figure 8.2. (c) The relation R 3 on A={a, b, c} given by R 3 = { (a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c) } = A 2 is shown bottom-left in figure 8.2. (d) The relation R 4 on A={a, b, c} given by R 4 = is shown bottom-right in figure 8.2. There are many things we can do with a graphical representation of a relation and in fact the theory of graphs will take up a whole chapter later. But for now they are mainly useful as a way of checking which properties a given relation has or does not have Properties of relations There are seven properties which allow us to distinguish some particularly important kinds of relations. It will be convenient to present them in a table. This

12 56 CHAPTER 8. CARTESIAN PRODUCT AND RELATIONS is given below as table 8.1: each row gives the formal definition of the property and then how it may be recognised in the graph of the relation. We can check the Property Definition Trivial R= no edges Universal R=A 2 an edge in both directions between any pair of vertices+aloop at every vertex Total for any a and b in A, a b, (a, b) R (b, a) R an edge in at least one direction between any pair of vertices Reflexive for every a A, (a, a) R Symmetric Antisymmetric a loop at every vertex: for any a and b in A, if (a, b) R then (b, a) R any edge matched by one in the opposite direction: for any a and b in A, if (a, b) R and (b, a) Rthen a=b no edge matched by the opposite edge: Transitive for any a, b and c in A, if (a, b) R and (b, c) R then (a, c) R consecutive edges must complete a triangle Table 8.1: the seven key properties for a relation R on set A. relations shown graphically in figure 8.2: R 1 Trivial Universal Total Reflexive Symmetric Antisymmetric Transitive R 2 R 3 R 4 You should confirm that you agree with all the ticks! In particular, notice that R 4, the trivial relation, is simultaneously symmetric and antisymmetric! All its edges satisfy both conditions because it has no edges to fail! Mathematicians say the properties are satisfied vacuously. The most subtle property is transitivity. In R 1 we have some consecutive edges which form part of a triangle: (b, c) and (c, d) have the edge (b, d), for example. But (a, b) and (b, d) do not have (a, d). There is a triangle with edge (d, a) but not the right kind of triangle. And for (a, b) and (b, c) there is no triangle at all, not even one of the wrong kind.

13 8.2. RELATIONS Special types of relations The reason for distinguishing the seven properties in the last section is that they combine to distinguish some special types of relations which occur everywhere in mathematics and computer science, from relational databases to algebraic geometry. Equivalence Relation R is an equivalence relation, or an equivalence, if it is reflexive, symmetric and transitive; Partial Order R is a partial order if it is reflexive, antisymmetric and transitive; Total Order R is a total order if it is total and is a partial order. We can summarise these in a table too: Equivalence Partial order Total order Trivial Universal Total Reflexive Symmetric Antisymmetric Transitive If we look back to the table which summarised the properties of the graphs in figure 8.2 then we see that R 2 is a total order and R 3 is an equivalence relation. It is not a coincidence that the equivalence relation, R 3, also happened to be universal. We have the following, very important fact: Theorem 23 Any equivalence relation R over a set A is made up of a union of disjoint universal relations which together include all elements of A. In symbols, there is a possibly infinite list of disjoint subsets A 1, A 2,... of A for which 1. A 1 A 2...=A (the A i are said to partition A); 2. A 2 1 A2 2...=R. The sets A i in Theorem 23 are called the equivalence classes of the relation. E.g. (a) The relation R on A={a, b, c, d, e, f} given by R 1 = { (a, a), (a, b), (a, d), (b, a), (b, b), (b, d), (d, a), (d, b), (d, d), (c, c), (c, f ), ( f, c), ( f, f ), (e, e) } is an equivalence relation with equivalence classes A 1 ={a, b, d}, A 2 ={c, f} and A 3 ={e}. Its graph is shown in figure 8.3. (b) The relation R on the integerszgiven by

14 58 CHAPTER 8. CARTESIAN PRODUCT AND RELATIONS R = { (x, y) x, y Z x and y have the same parity (both odd or both even) } is an equivalence relation with equivalence classes: A 1 ={..., 5, 3, 1, 1, 3, 5, 7,...}, (the odd numbers) A 2 ={..., 4, 2, 0, 2, 4, 6,...}, (the even numbers). We can think of these equivalence classes as being the same as the elements ofz 2, the integers modulo 2. (c) The relation R on the positive integersz >0 given by R = { (x, y) x, y Z >0 the highest power of 2 dividing x and y is the same } is an equivalence relation with equivalence classes: A 1 ={1, 3, 5, 7,...}, A 2 ={2 1, 2 3, 2 5,...}, A 3 ={2 2 1, 2 2 3, 2 2 5,...}... and so on an infinite list of infinite equivalence classes! Figure 8.3: an equivalence relation on A ={a, b, c, d, e, f} consisting of 3 disjoint universal relations and partitioning A into three equivalence classes. Partial orders have a theorem too. We know that in a partial order cyclic paths of length two are banned because of antisymmetry (see row 6 of table 8.1). What about cyclic paths of length three: (a, b), (b, c) and (c, a)? Well, a partial order has to be transitive, so (a, b), (b, c) requires that (a, c) is in the relation. But now we have (c, a) and (a, c) which is not allowed by antisymmetry. So there are no cyclic triangles. In a similar way, we see that: Theorem 24 Apart from loops, in the graph of a partial order there can be no cyclic paths. As a consequence of this theorem, we can draw the graph of a partial order so that all the edges point in the same direction. It is traditional to make them point upwards. Then we leave the arrows off since we know which direction they point (upwards). Then we leave off the loops since we know that every vertex has a loop because of reflexivity. Then we leave out all the triangle completions because we know they happen by transitivity. The result is a much simpler graph

15 8.2. RELATIONS 59 which still has all the information about the partial order. It is called a Hasse diagram. Figure 8.4 shows the derivation of the Hasse diagram for the relation R 2 in figure 8.2. Figure 8.4: deriving the Hasse diagram of the relation R 2 from figure 8.2.