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CS2013: Relations and Functions

CS2013: Relations and Functions. Relations and Functions. Some background for CS2013 Necessary for understanding the difference between Deterministic FSAs (DFSAs) and NonDeterministic FSAs (NDFSAs). If what follows is new or puzzling….

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CS2013: Relations and Functions

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  1. CS2013: Relations and Functions Kees van Deemter

  2. Relations and Functions • Some background for CS2013 • Necessary for understanding the difference between • Deterministic FSAs (DFSAs) and • NonDeterministic FSAs (NDFSAs) Kees van Deemter

  3. If what follows is new or puzzling… • … then read K.H.Rosen, Discrete Mathematics and Its Applications, theChapters on sets, functions, and relations (Chapters 2 and 9 in the 7th edition). • Free copies in pdf can be found on the web http://www2.fiit.stuba.sk/~kvasnicka/Mathematics%20for%20Informatics/Rosen_Discrete_Mathematics_and_Its_Applications_7th_Edition.pdf Kees van Deemter

  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 Kees van Deemter

  5. Cartesian product • The Cartesian product of n sets A1 x A2 x … x An 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) Kees van Deemter

  6. Student x CAS = {(John,1),(John,2),… (John,20),(Mary,1),…,(Mary,20)} • A1 x A2 x … x An = {(x1,x2,…,xn): x1A and x2A and … and xnA} Kees van Deemter

  7. Binary Relations • Let A, B be sets. A binary relationR 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 thata is related to b (by R) • Also written asaRb; alsoR(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} Kees van Deemter

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

  9. Inverse Relations Any binary relation R:A×B has an inverse relation R−1:B×A, defined byR−1 :≡ {(b,a) | (a,b)R}. E.g., <−1 = {(b,a) | a<b} = {(b,a) | b>a} = > Kees van Deemter

  10. Reflexivity and relatives • A relation R on A is reflexiveiff 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 “notreflexive”, 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 Kees van Deemter

  11. Some examples • Reflexive: =, ‘have same cardinality’,  <=, >=, , , etc. Kees van Deemter

  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? Kees van Deemter

  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) Kees van Deemter

  14. Antisymmetry • Consider the relation xy • Is it symmetrical? • Is it asymmetrical? • Is it reflexive? • Is it irreflexive? Kees van Deemter

  15. Antisymmetry • Consider the relation xy • Is it symmetrical? No • Is it asymmetrical? • Is it reflexive? • Is it irreflexive? Kees van Deemter

  16. Antisymmetry • Consider the relation xy • Is it symmetrical? No • Is it asymmetrical? No • Is it reflexive? • Is it irreflexive? Kees van Deemter

  17. Antisymmetry • Consider the relation xy • Is it symmetrical? No • Is it asymmetrical? No • Is it reflexive? Yes • Is it irreflexive? Kees van Deemter

  18. Antisymmetry • Consider the relation xy • Is it symmetrical? No • Is it asymmetrical? No • Is it reflexive? Yes • Is it irreflexive? No Kees van Deemter

  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 Kees van Deemter

  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)} Kees van Deemter

  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. Kees van Deemter

  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” Kees van Deemter

  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 Kees van Deemter

  24. End of aside Kees van Deemter

  25. the difference between relations and functions Totality: • 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). Kees van Deemter

  26. Functionality Functionality: • A relation R: A×B is functionaliff, 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)). Kees van Deemter

  27. Functionality • R: A×B is functionaliff, for every aA, there is at most one bB such that (a,b)R. aA: ¬ b1,b2 B (b1≠b2  aRb1  aRb2). • 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. Kees van Deemter

  28. Examples • Consider the relation Scored again: • A relation between Student and CAS • Is it a total relation? • Is it a functional relation? Kees van Deemter

  29. Functionality for 3-place relations • Consider a 3-place relation R • R is a subset of A1 x A2 x A3, (for some A1, A2, A3) • R is functional in its first two arguments iffor all xA1 and yA2, there exists at most one zA3 such that (x,y,z)  R. • This is easy to generalise to n arguments Kees van Deemter

  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. Kees van Deemter

  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? Kees van Deemter

  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 Kees van Deemter

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