1. Relations
Rosen 5th ed., ch. 7

2. Relations Relationships between elements of sets occur very often.
(Employee, Salary)
(Students, Courses, GPA)
We use ordered pairs (or n-tuples) of elements from the sets to represent relationships.

3. Binary Relations
A binary relation R from set A to set B, written R:A↔B, is a subset of A×B.
E.g., let < : N↔N :≡ {(n,m) | n < m}
The notation aRb means (a,b)R.
E.g., a < b means (a,b) <
If aRb we may say
“a is related to b (by relation R)”

4. Example A: {students at UNR}, B: {courses offered at UNR}
R: “relation of students enrolled in courses”
(Jason, CS365), (Mary, CS201) are in R
If Mary does not take CS365, then (Mary, CS365) is not in R!
If CS480 is not being offered, then (Jason, CS480), (Mary, CS480) are not in R!

5. Complementary Relations
Let R:A↔B be any binary relation.
Then, R:A↔B, the complement of R, is the binary relation defined by R :≡ {(a,b) | (a,b)R} = (A×B) − R
Note this is just R if the universe of discourse is U = A×B; thus the name complement.
Note the complement of R is R.
Example: < = {(a,b) | (a,b)<} = {(a,b) | ¬a<b} = ≥

6. 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} = >.

7. Functions as Relations
A function f:AB is a relation from A to B
A relation from A to B is not always a function f:AB (e.g., relations could be one-to-many)
Relations are generalizations of functions!

8. Relations on a Set
A (binary) relation from a set A to itself is called a relation on the set A.
A: {1,2,3,4}
R={(a,b) | a divides b}

9. Example
How many relations are there on a set A with n elements?

10. Reflexivity A relation R on A is reflexive if aA, aRa.
E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive.
Is the “divide” relation on the set of positive integers reflexive?

11. Symmetry & Antisymmetry
A binary relation R on A is symmetric iff R = R−1, that is, if (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 antisymmetric if (a,b)R → (b,a)R.
< is antisymmetric, “likes” is not.

12. Transitivity
A relation R is transitive iff (for all a,b,c) (a,b)R  (b,c)R → (a,c)R.
Examples:
“is an ancestor of” is transitive.
Is the “divides” relation on the set of positive integers transitive?

13. Combining Relations
Two relations can be combined in a similar way to combining two sets.

14. Composite Relations
Let R:A↔B, and S:B↔C. Then the composite SR of R and S is defined as:
SR = {(a,c) | aRb  bSc}
Function composition fg is an example.
The nth power Rn of a relation R on a set A can be defined recursively by:
R1 :≡ R ; Rn+1 :≡ RnR for all n≥0.

15. Example RS = {(1,0),(1,1),(2,1),(2,2),(3,0),(3,1)}
R is a relation from {1,2,3} to {1,2,3,4}
R ={(1,1),(1,4),(2,3),(3,1),(3,4)}
S is a relation from {1,2,3,4} to {0,1,2}
S = {(1,0),(2,0),(3,1),(3,2),(4,1)}
RS = {(1,0),(1,1),(2,1),(2,2),(3,0),(3,1)}

16. §7.2: n-ary Relations
An n-ary relation R on sets A1,…,An, written R:A1,…,An, is a subset R  A1× … × An.
The degree of R is n.
Example: R consists of 5-tuples (A,N,S,D,T)
A: airplane flights, N: flight number,
S: starting point, D: destination, T: departure time

17. Databases
The time required to manipulate information in a database depends in how this information is stored.
Operations: add/delete, update, search, combine etc.
Various methods for representing databases have been developed.
We will discuss the “relational model”.

18. Relational Databases
A database consists of records, which are n-tuples, made up of fields.
A relational database represents records as an n-ary relation R.
(STUDENT_NAME, ID, MAJOR, GPA)
Relations are also called “tables” (e.g., displayed as tables often)

19. Relational Databases
A domain Ai of an n-ary relation is called primary key when no two n-tuples have the same value on this domain (e.g., ID)
A composite key is a subset of domains {Ai, Aj, …} such that an n-tuple (…,ai,…,aj,…) is determined uniquely for each composite value (ai, aj,…)Ai×Aj×…

20. Selection Operator
Let R be an n-ary relation and C a condition that elements in R may satisfy.
The selection operator sC maps any (n-ary) relation R on A to the n-ary relation of all n-tuples from R that satisfy C.

21. Example
Suppose we have a domain A = StudentName × Standing × SocSecNos
Suppose we define a certain condition on A, UpperLevel(name,standing,ssn) :≡ [(standing = junior)  (standing = senior)]
Then, sUpperLevel is the selection operator that takes any relation R on A (database of students) and produces a relation consisting of just the upper-level classes (juniors and seniors).

22. Projection Operators The projection operator maps the n-tuple
to the m-tuple
In other words, it deletes n-m of the components of an n-tuple.

23. Example
Note that fewer rows may result when a projection is applied !

24. More Examples
Suppose we have a ternary (3-ary) domain Cars=Model×Year×Color. (note n=3).
Consider the index sequence {ik}= 1,3. (m=2)
Then the projection P simply maps each tuple (a1,a2,a3) = (model,year,color) to its image:
This operator can be usefully applied to a whole relation RCars (database of cars) to obtain a list of model/color combinations available.
{ik}

25. Join Operator
Puts two relations together to form a sort of combined relation.
If the tuple (A,B) appears in R1, and the tuple (B,C) appears in R2, then the tuple (A,B,C) appears in the join J(R1,R2).
A, B, C can also be sequences of elements rather than single elements.

26. Example

27. Join Example
Suppose R1 is a teaching assignment table, relating Professors to Courses.
Suppose R2 is a room assignment table relating Courses to Rooms,Times.
Then J(R1,R2) is like your class schedule, listing (professor,course,room,time).

28. SQL Example SELECT Departure_Time FROM Flights
WHERE destination=“Detroit”
Find projection P5 of the selection of 5-tuples that satisfy the constraint “destination=Detroit”

29. §7.3: Representing Relations
Some ways to represent n-ary relations:
With an explicit list or table of its tuples.
With a function from the domain to {T,F}.
Or with an algorithm for computing this function.
Some special ways to represent binary relations:
With a zero-one matrix.
With a directed graph.

30. Using Zero-One Matrices
To represent a relation R by a matrix MR = [mij], let mij = 1 if (ai,bj)R, else 0.
E.g., Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally.
The 0-1 matrix representation of that “Likes” relation:

31. Zero-One Reflexive, Symmetric
Terms: Reflexive, non-Reflexive, symmetric, and antisymmetric.
These relation characteristics are very easy to recognize by inspection of the zero-one matrix.
any- thing
any- thing
anything
anything
any- thing
any- thing
Reflexive: all 1’s on diagonal
Non-reflexive: some 0’s on diagonal
Symmetric: all identical across diagonal
Antisymmetric: all 1’s are across from 0’s

32. Using Directed Graphs
A directed graph or digraph G=(VG,EG) is a set VG of vertices (nodes) with a set EGVG×VG of edges (arcs,links). Visually represented using dots for nodes, and arrows for edges. Notice that a relation R:A↔B can be represented as a graph GR=(VG=AB, EG=R).
Edge set EG (blue arrows) MR GR
Joe
Susan
Fred
Mary
Mark
Sally
Node set VG (black dots)

33. Digraph Reflexive, Symmetric
It is extremely easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.           
Reflexive: Every node has a self-loop
Irreflexive: No node links to itself
Symmetric: Every link is bidirectional
Antisymmetric: No link is bidirectional
Asymmetric, non-antisymmetric
Non-reflexive, non-irreflexive

34. §7.4: Closures of Relations
For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property.
The reflexive closure of a relation R on A is obtained by adding (a,a) to R for each aA. I.e., it is R  IA
The symmetric closure of R is obtained by adding (b,a) to R for each (a,b) in R. I.e., it is R  R−1
The transitive closure or connectivity relation of R is obtained by repeatedly adding (a,c) to R for each (a,b),(b,c) in R.
I.e., it is

35. Paths in Digraphs/Binary Relations
A path of length n from node a to b in the directed graph G (or the binary relation R) is a sequence (a,x1), (x1,x2), …, (xn−1,b) of n ordered pairs in EG (or R).
An empty sequence of edges is considered a path of length 0 from a to a.
If any path from a to b exists, then we say that a is connected to b. (“You can get there from here.”)
A path of length n≥1 from a to a is called a circuit or a cycle.
Note that there exists a path of length n from a to b in R if and only if (a,b)Rn.

36. Simple Transitive Closure Alg.
A procedure to compute R* with 0-1 matrices.
procedure transClosure(MR:rank-n 0-1 mat.)
A := B := MR;
for i := 2 to n begin A := A⊙MR; B := B  A {join} end return B {Alg. takes Θ(n4) time}
{note A represents Ri}

37. Roy-Warshall Algorithm
Uses only Θ(n3) operations!
Procedure Warshall(MR : rank-n 0-1 matrix)
W := MR
for k := 1 to n for i := 1 to n for j := 1 to n wij := wij  (wik  wkj) return W {this represents R*}
wij = 1 means there is a path from i to j going only through nodes ≤k

38. §7.5: Equivalence Relations
An equivalence relation (e.r.) on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive.
E.g., = itself is an equivalence relation.
For any function f:A→B, the relation “have the same f value”, or =f :≡ {(a1,a2) | f(a1)=f(a2)} is an equivalence relation, e.g., let m=“mother of” then =m = “have the same mother” is an e.r.

39. Equivalence Relation Examples
“Strings a and b are the same length.”
“Integers a and b have the same absolute value.”
“Real numbers a and b have the same fractional part (i.e., a − b  Z).”

40. Equivalence Classes Let R be any equiv. rel. on a set A.
The equivalence class of a, [a]R :≡ { b | aRb } (optional subscript R)
It is the set of all elements of A that are “equivalent” to a according to the eq.rel. R.
Each such b (including a itself) is called a representative of [a]R.
Since f(a)=[a]R is a function of a, any equivalence relation R be defined using aRb :≡ “a and b have the same f value”, given that f.

41. Equivalence Class Examples
“Strings a and b are the same length.”
[a] = the set of all strings of the same length as a.
“Integers a and b have the same absolute value.”
[a] = the set {a, −a}
“Real numbers a and b have the same fractional part (i.e., a − b  Z).”
[a] = the set {…, a−2, a−1, a, a+1, a+2, …}

42. Partitions
A partition of a set A is the set of all the equivalence classes {A1, A2, … } for some e.r. on A.
The Ai’s are all disjoint and their union = A.
They “partition” the set into pieces. Within each piece, all members of the set are equivalent to each other.