CS2013: Relations and Functions

Transcription

1 CS2013: Relations and Functions 20/09/16 Kees van Deemter 1

2 Relations and Functions Some background for CS2013 Necessary for understanding the difference between Deterministic FSAs (DFSAs) and NonDeterministic FSAs (NDFSAs) 20/09/16 Kees van Deemter 2

3 If what follows is new or puzzling then read K.H.Rosen, Discrete Mathematics and Its Applications, the Chapters on sets, functions, and relations (Chapters 2 and 9 in the 7 th edition). Free copies in pdf can be found on the web Rosen_Discrete_Mathematics_and_Its_Applications_7th_Edition.pdf 20/09/16 Kees van Deemter 3

4 Relations and Functions Simple mathematical constructs Based on elementary set theory Can be used to model many things including the set of edges in a given FSA 20/09/16 Kees van Deemter 4

5 Cartesian product The Cartesian product of n sets A 1 x A 2 x x A n First n=2 (a 2-place relation) A x B = The Cartesian product of A and B = {(x,y): x A and y B}. Example: A= set of all students (e.g., John, Mary), B=set of all CAS marks (e.g., 1-20) 20/09/16 Kees van Deemter 5

6 Student x CAS = {(John,1),(John,2), (John, 20),(Mary,1),,(Mary,20)} A 1 x A 2 x x A n = {(x 1,x 2,,x n ): x 1 A and x 2 A and and x n A} 20/09/16 Kees van Deemter 6

7 Binary Relations Let A, B be sets. A binary relation R from A to B is a subset of A B. Analogous for n-ary relations E.g., < can be seen as {(n,m) n < m} (a,b) R means that a is related to b (by R) Also written as arb; also R(a,b) Can be used to model real-life facts. E.g., Scored = {(x Student,y CAS): x scored y in last years s CS2013 exam} 20/09/16 Kees van Deemter 7

8 Binary Relations Aside: This way of modelling relations using sets suggests some natural questions and operations, e.g., 20/09/16 Kees van Deemter 8

9 Inverse Relations Any binary relation R:A B has an inverse relation R 1 :B A, defined by R 1 : {(b,a) (a,b) R}. E.g., < 1 = {(b,a) a<b} = {(b,a) b>a} = > 20/09/16 Kees van Deemter 9

10 Reflexivity and relatives A relation R on A is reflexive iff a A(aRa). E.g., the relation : {(a,b) a b} is reflexive. R is irreflexive iff a A( ara) Note irreflexive does NOT mean not reflexive, which is just a A(aRa). E.g., if Adore={(j,m),(b,m),(m,b)(j,j)} then this relation is neither reflexive nor irreflexive 20/09/16 Kees van Deemter 10

11 Some examples Reflexive: =, have same cardinality, <=, >=,,, etc. 20/09/16 Kees van Deemter 11

12 Symmetry & relatives A binary relation R on A is symmetric iff a,b((a,b) R (b,a) R). E.g., = (equality) is symmetric. < is not. is married to is symmetric, likes is not. A binary relation R is asymmetric if a,b((a,b) R (b,a) R). Examples: < is asymmetric, Adores is not. Let R={(j,m),(b,m),(j,j)}. Is R (a)symmetric? 20/09/16 Kees van Deemter 12

13 Symmetry & relatives Let R={(j,m),(b,m),(j,j)}. R is not symmetric (because it does not contain (m,b) and because it does not contain (m,j)). R is not asymmetric, due to (j,j) 20/09/16 Kees van Deemter 13

14 Antisymmetry Consider the relation x y Is it symmetrical? Is it asymmetrical? Is it reflexive? Is it irreflexive? 20/09/16 Kees van Deemter 14

15 Antisymmetry Consider the relation x y Is it symmetrical? No Is it asymmetrical? Is it reflexive? Is it irreflexive? 20/09/16 Kees van Deemter 15

16 Antisymmetry Consider the relation x y Is it symmetrical? No Is it asymmetrical? No Is it reflexive? Is it irreflexive? 20/09/16 Kees van Deemter 16

17 Antisymmetry Consider the relation x y Is it symmetrical? No Is it asymmetrical? No Is it reflexive? Yes Is it irreflexive? 20/09/16 Kees van Deemter 17

18 Antisymmetry Consider the relation x y Is it symmetrical? No Is it asymmetrical? No Is it reflexive? Yes Is it irreflexive? No 20/09/16 Kees van Deemter 18

19 Antisymmetry Consider the relation x y It is not symmetric. (For instance, 5 6 but not 6 5) It is not asymmetric. (For instance, 5 5) The pattern: the only times when (a,b) and (b,a) are when a=b This is called antisymmetry 20/09/16 Kees van Deemter 19

20 Antisymmetry A binary relation R on A is antisymmetric iff a,b((a,b) R (b,a) R) a=b). Examples:,, Another example: the earlier-defined relation Adore={(j,m),(b,m),(j,j)} 20/09/16 Kees van Deemter 20

21 Transitivity & relatives A relation R is transitive iff (for all a,b,c) ((a,b) R (b,c) R) (a,c) R. A relation is nontransitive iff it is not transitive. A relation R is intransitive iff (for all a,b,c) ((a,b) R (b,c) R) (a,c) R. 20/09/16 Kees van Deemter 21

22 Transitivity & relatives What about these examples: x is an ancestor of y x likes y x is located within 1 mile of y x +1 =y x beat y in the tournament 20/09/16 Kees van Deemter 22

23 Transitivity & relatives What about these examples: is an ancestor of is transitive. likes is neither trans nor intrans. is located within 1 mile of is neither trans nor intrans x +1 =y is intransitive x beat y in the tournament is neither trans nor intrans 20/09/16 Kees van Deemter 23

24 End of aside 20/09/16 Kees van Deemter 24

25 Totality: the difference between relations and functions A relation R:A B is total if for every a A, there is at least one b B such that (a,b) R. N.B., it does not follow that R 1 is total It does not follow that R is functional (see over). 20/09/16 Kees van Deemter 25

26 Functionality Functionality: A relation R: A B is functional iff, for every a A, there is at most one b B such that (a,b) R. A functional relation R: A B does not have to be total (there may be a A such that b B (arb)). 20/09/16 Kees van Deemter 26

27 Functionality R: A B is functional iff, for every a A, there is at most one b B such that (a,b) R. a A: b 1, b 2 B (b 1 b 2 arb 1 arb 2 ). If R is a functional and total relation, then R can be seen as a function R: A B Hence one can write R(a)=b as well as arb, R(a,b), and (a,b) R. Each of these mean the same. 20/09/16 Kees van Deemter 27

28 Examples Consider the relation Scored again: A relation between Student and CAS Is it a total relation? Is it a functional relation? 20/09/16 Kees van Deemter 28

29 Functionality for 3-place relations Consider a 3-place relation R R is a subset of A 1 x A 2 x A 3, (for some A 1, A 2, A 3 ) R is functional in its first two arguments if for all x A 1 and y A 2, there exists at most one z A 3 such that (x,y,z) R. This is easy to generalise to n arguments 20/09/16 Kees van Deemter 29

30 Examples Suppose you model addition of natural numbers as a 3-place relation (0,0,0),(0,1,1), (1,0,1), (1,1,2), This relation is functional in its first two arguments. 20/09/16 Kees van Deemter 30

31 Examples Let Scored be a subset of Student x CAS x PASS, namely {(student,casmark,yes/no): student scored casmark and passed yes/no} Is the relation Scored functional in its first two arguments? 20/09/16 Kees van Deemter 31

32 Examples Let Scored be a subset of Student x CAS x PASS, namely {(student,casmark,yes/no): student scored casmark and passed yes/no} Is the relation Scored functional in its first two arguments? Yes: given (a student and) a CAS mark, you cannot have both pass-yes and pass-no 20/09/16 Kees van Deemter 32