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Discrete Maths

Discrete Maths. 242/240-213 , Semester 2, 2018 - 2019. Objective to introduce relations, show their connection to sets, and their use in databases. 5. Relations. Overview. Defining a Relation Relations using One Set Properties of a Relation reflexive, symmetric, transitive

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Discrete Maths

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  1. Discrete Maths 242/240-213, Semester 2,2018-2019 • Objective • to introduce relations, show their connection to sets, and their use in databases 5. Relations

  2. Overview • Defining a Relation • Relations using One Set • Properties of a Relation • reflexive, symmetric, transitive • Composition of Relations • N-ary Relations • Databases and Relations • More Information

  3. 1. Defining a Relation • A relation connects two (or more) sets. • Two new ideas: • the Cartesian Product (AB) of sets • ordered pairs

  4. Cartesian Product Example • A = {Smith, Johnson} • B = {Calc, Math, History, Programming} B A  Art Smith Math History Johnson Programming the Cartesian Product creates all the possible links from elements in set A to set B

  5. A Set of Ordered Pairs • We can write these links as a set of ordered pairs, one pair for each link: • AxB = { (Smith, Art), (Smith,Math), (Smith,History), ..., (Johnson,History), (Johnston,Programming) } • The ordering of a pair matters: first an element from A, which is linked to an element from B. • A pair is sometimes called a 2-tuple.

  6. Relations • A relation R is defined by a subset of the ordered pairs in AxB. • For example one possible relation is: B A R Art Smith Math History Johnson Programming

  7. R can be written as a set of ordered pairs: • R = { (Smith,Art), (Smith,Math), ... (Johnson, History), (Johnston, Programming) } • R  A x B • We can also write each link as a relation (or predicate): • (Smith R Art) is trueor R(Smith, Art) is true • (Johnston R Art) is falseor R(Johnston,Art) is false yes, there is a link between relations and predicate logic

  8. 2. Relations Using One Set • It's possible to use the same set in a relation • e.g. R  A x A • For example, integer relations are  Z x Z R A A : : -1 -1 0 0 1 1 2 2 : : b = a + 1

  9. A relation involving numerical sets can often be summarized using set notation. • For example: • R = {(a,b) | a  Z, b  Z, b = a+1 } or without the set types: • R = {(a,b) |b = a+1 }

  10. Relations in Python

  11. 3. Properties of a Relation A relation R involving a set S may have special properties: • If (x R x) is true, then Ris reflexive. • If (x R y) is true when (y R x) is true, then Ris symmetric. • If (x R z) is true when (x R y) and (y R z) are true, then Ris transitive. S S R x x R x y y x R x y R R x y z y z R

  12. Properties in Words • Reflexive : "all go to self" • Symmetric: "arrows go both ways" • Transitive: "things which are linked, are linked with one arrow"

  13. Properties as Drawings • If the two sets are the same in a relation (R  S x S), then you can draw the relations as arrows between the things. • e.g. S = {1,2,3,4}, R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)} 1 2 R S S 1 1 2 2 OR 3 3 3 4 4 4

  14. Relation Property Examples • S = {0,1,2,3} R = {(0,3), (2, 3)} Reflexive: no, e.g. no (0,0) Symmetric: no, e.g. no (3,0) Transitive: yes • S = {0,1,2,3} R = {(0,0), (0,1), (0,2), (1,3)} Reflexive: no, e.g. no (1,1) Symmetric: no, e.g. no (3,1) Transitive: no, (0,1) and (1,3) but no (0,3) 0 1 2 3 0 1 2 3

  15. S = {0,1,2,3} R = {(1,2), (2,1), (1,3), (3,1)} Reflexive: no, e.g. no (0,0) Symmetric: yes Transitive: no, (2,1) and (1,3) but no (2,3) • S = {0,1,2,3} R = {(0,0), (1,1), (2,2), (0,1), (1,2), (2,3)} Reflexive: no, e.g. no (3,3) Symmetric: no, e.g. no (2,1) Transitive: no, (0,1) and (1,2) but no (0,2) 0 1 2 3 0 1 2 3

  16. Not Examinable Python Code for Property Testing def isReflexive(R): # all go to self elems = set() for (x,y) in R: # collect all elements elems.add(x) elems.add(y) for el in elems: if (el,el) not in R: return False return True def isSymmetric(R): # all arrows go both ways for (x,y) in R: if (y,x) not in R: return False return True def isTransitive(R): # all things which are linked, # are linked with one arrow for (x,y) in R: for (y1,z) in R: if (y == y1): # x --> y --> z if (x,z) not in R: return False return True def test(R): # test relation for three properties print("R:", R) print("-Reflexive", isReflexive(R)) print("-Symmetric", isSymmetric(R)) print("-Transitive", isTransitive(R)) print("")

  17. Examples print ("- 1 ----------------------") test({(0,3), (2, 3)}) test({(0,0), (0,1), (0,2), (1,3)}) test({(1,2), (2,1), (1,3), (3,1)}) test({(0,0), (1,1), (2,2), (0,1), (1,2), (2,3)}) print ("- 2 ----------------------") test({ (1,1), (2,2), (3,3), (2,1) }) test({ (1,2), (2,1), (3,3) }) test({ (0,1),(1,2),(2,3),(0,2),(1,3),(0,3) }) print ("- 3 -----------------------") nums = {1,2,3,4} test({(x,y) for x in nums for y in nums }) print ("- 4 -----------------------") test({(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}) test({(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}) test({(2,4), (4, 2)}) print ("- 5 -----------------------") test({(1, 2), (2, 3), (3, 4)}) test({(1,1),(2,2),(3,3),(4,4)}) test({(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)})

  18. Make sure you understand why the results are true/false. - 2 ---------------------- R: {(3, 3), (1, 1), (2, 1), (2, 2)} -Reflexive True -Symmetric False -Transitive True R: {(1, 2), (3, 3), (2, 1)} -Reflexive False -Symmetric True -Transitive False R: {(0, 1), (1, 2), (1, 3), (2, 3), (0, 3), (0, 2)} -Reflexive False -Symmetric False -Transitive True - 3 ----------------------- R: {(1, 2), (3, 2), (1, 3), (4, 1), (3, 3), (3, 1), (4, 4), (2, 1), (1, 4), (2, 4), (2, 3), (4, 3), (2, 2), (4, 2), (3, 4), (1, 1)} -Reflexive True -Symmetric True -Transitive True - 1 ---------------------- R: {(0, 3), (2, 3)} -Reflexive False -Symmetric False -Transitive True R: {(0, 1), (0, 0), (0, 2), (1, 3)} -Reflexive False -Symmetric False -Transitive False R: {(1, 2), (1, 3), (3, 1), (2, 1)} -Reflexive False -Symmetric True -Transitive False R: {(0, 1), (1, 2), (0, 0), (2, 3), (2, 2), (1, 1)} -Reflexive False -Symmetric False -Transitive False

  19. Make sure you understand why the results are true/false. - 5 ----------------------- R: {(1, 2), (3, 4), (2, 3)} -Reflexive False -Symmetric False -Transitive False R: {(3, 3), (4, 4), (1, 1), (2, 2)} -Reflexive True -Symmetric True -Transitive True R: {(1, 3), (3, 1), (1, 4), (2, 3), (3, 4), (2, 4)} -Reflexive False -Symmetric False -Transitive False - 4 ----------------------- R: {(3, 2), (3, 3), (2, 3), (2, 2), (3, 4), (2, 4)} -Reflexive False -Symmetric False -Transitive True R: {(1, 2), (3, 3), (4, 4), (2, 1), (2, 2), (1, 1)} -Reflexive True -Symmetric True -Transitive True R: {(4, 2), (2, 4)} -Reflexive False -Symmetric True -Transitive False

  20. 4. Composition of Relations • Let R be a relation from a set A to a set B and S a relation from B to a set C. • The composite of R and S is written as SR • read this as "do R then do S" this strange ordering is to do with the connection between relations and functions (see the next part) R S a s 9 2 t 0 : : : d 3 h 4 4 u A B C

  21. Example 1 • Let R and S be relations on Z+= {1, 2, 3, …} R = {(a, b) | b = 2*a} S = {(a, b) | b = a-1} R S 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 : : : Z+ Z+ Z+

  22. We can write SR in terms of the ordered pairs of "start" and "finish" elements: • SR = {(1,1), (2,3), (3,5), (4,7), ... } • remember that  means "do R and then do S" • We can also summarise the composite using set notation: SR = {(a,b) | b = (2*a) – 1 } this combines the maths of set R and S

  23. Example 2 • Let R and S be relations on A = {1, 2, 3, 4} • R = {(a, b) | b = 5 - a} • S = {(a, b) | b > a} • this means that an 'a' can be linked to all 'b's which are bigger S 1 1 1 R 2 2 2 3 3 3 4 4 4 A A A

  24. As a set of ordered pairs: • SR = { (2,4), (3,3), (3,4), (4,2), (4,3), (4,4) } • only "start" and "finish" pairs are included • Using set notation: • S°R = {(a,b) | b > 5 – a} or • S°R = {(a,b) | a + b > 5} this combines the maths of set R and S

  25. 5. N-ary Relations • The Cartesian Product can involve any number of sets. We write it as A1 x A2 x … x An. • The sets A1, A2, …, An are called the domains of the relation, and n is called its degree. for this example, degree == 3 a s 9 2 0 4 h 4 u A1 A2 A3

  26. An n-ary relation R uses a subset of the links in an n-ary Cartesian Product. • e.g. a s 9 2 0 4 h 4 u A1 A2 A3

  27. 6. Databases and Relations • A database can be defined as an n-ary relation • this is known as therelational data model • A 'database' for students: • R = { (Ackermann, 231, CoE, 3.88), (Adams, 888, Physics, 3.45), (Chou, 102, CoE, 3.79), (Goodfriend, 453, Math, 3.45), (Rao, 678, Math, 3.90), (Stevens, 786, Math, 2.99) } • Each tuple is also called a record. called a 4-tuple (because there are 4 values)

  28. Each tuple (record) is made up of values from sets (also called domainsor fields). • e.g. each R tuple contains values from the fields Name, ID, Dept, and GPA • The database R is an n-ary relation: 001 0.01 CoE 002 0.02 Ackerman : : Math Adams 231 3.45 : : Physics Chou 888 3.88 = 1 tuple(or record) Chem : : : : : 4.00 Name ID Dept GPA 4 fields (domains) primary key

  29. Primary Key • A field is called a primary key if the relation's tuples are uniquely defined by that key's values • no two records can have the same primary key value • e.g. ID is the primary key in the R relation. • e.g. If you know ID == 102 then you know which tuple is wanted. • But if you know Dept == Math then you don't know which tuple is wanted.

  30. 7. More Information • Discrete Mathematics and its ApplicationsKenneth H. RosenMcGraw Hill, 2007, 7th edition • chapter 9, sections 9.1 – 9.2

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