1. CS2013: Relations and Functions
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Kees van Deemter

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

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 7th edition).
Free copies in pdf can be found on the web
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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
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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)
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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}
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Kees van Deemter

7. Binary Relations E.g., < can be seen as {(n,m) | n < m}
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}
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Kees van Deemter

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

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} = >
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Kees van Deemter

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
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Kees van Deemter

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

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

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

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

19. Antisymmetry Consider the relation xy This is called antisymmetry
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
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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)}
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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.
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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”
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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
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Kees van Deemter

24. End of aside
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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).
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Kees van Deemter

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)).
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Kees van Deemter

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: ¬ 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.
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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?
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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 if for 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
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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.
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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?
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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
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Kees van Deemter