Relations

1
Relations

• Two Views/Definitions of a Relation
• a) A set of n-tuples satisfying a certain n-ary
  predicate defines a relation, R, among the
  n-tuples.
• b) An n-ary relation, R, is a set of n-tuples
  that satisfy an n-ary predicate
• Just providing a set of n-tuples as the first
  definition suggests does not easily allow us to
  see the relation, R.
• e.g. (2,3), (3,2), (-2,-3), (-3,-2),
  (1,6),(6,1), (-1,-6),(-6,-1)
• Defining the relation, R, makes it a lot clearer.
• e.g. a,b integers I ab 6 (a,b)
• Recall that in predicates, the order of the
  elements is important. Similarly, the order of
  the elements is important in a relation.

2
Binary Relation

• A relation, R, which has only two elements is
  called a binary relation, and it is often the
  focus in this course.
• A binary relation, R, is a set of ordered pair
  satisfying a binary predicate.
• Example
• a,b integers I altb (a,b)
• this is the infinite set (-3,-2), (0,2), - - -
  - , (5,9) - - - -
• An ordered pair, (x,y), has the following
  characteristic
• (x,y) (m,n) ? x m and y n

3
Expressing Binary Relation

• There are several ways to express a binary
  relation, R
• x R y
• (x,y) R
• R(x,y).
• So far, we have demonstrated relation with
  numbers. The elements for a relation may come
  from any set.
• e.g. x R y allows x to come from set S1, and y
  to be from another set S2. (S1 may be a set of
  teachers, S2 may be a set of students, and R is
  the relationship of teaches.)

4
Cross Product

• A relation extracted from two sets, M and N, is
  usually referred to be a relation over M x N.
• M x N is known as the cross product or Cartesian
  product of M and N.
• The sets M and N may be the same. That is, we can
  have M x M as the cross product
• Example of a Relation over a Cross Product
• e.g. Let a set of doctors, D d1, d2, a set of
  patients, P p1, p2, p3, p4, p5 and let
  treats to be the relation, T. Then
• D x P (d1,p1), (d1, p2), (d1,p3), (d1,p4),
  (d1,p5), (d2,p1), (d2,p2), (d2,p3), (d2,p4),
  (d2,p5)
• T is a binary relation, treats, defined over D
  x P , where T (d1,p1), (d1,p3), (d2, p1),
  (d2,p2), (d2,p4), d2,p5)
• Relation T states that doctor d1 treats patients
  p1 and p3, and doctor d2 treats patients p1, p2,
  p4 and p5. Note that p1 is treated by both d1 and
  d2.

5
More interesting Example of Relation

• Consider a student record that contains a
  4-tuple name, age, classification, GPA (n, a,
  c, g)
• (n, a, c, g) name x age x classification x
  GPA
• Consider the relation, At_SPSU
• define At_SPSU relation to be the set
• At_SPSU (Jorge, 23, grad, 3.2)
  (Jill, 19,soph, 3.7) (Chaudary, 18, fresh,
  2.7) (Mike, 21, sen, 2.8) (Wang, 22, grad,
  3.4)

6
More interesting Example of Relation (cont.)

• Putting AT_SPSU relation in a tabular form as
  follows
• AT_SPSU
  Table

name
classif.
age
GPA
grad
Jorge
23
3.2
19
soph
Jill
3.7
Chaudary
18
fresh
2.7
21
sen
Mike
2.8
Wang
22
3.4
grad
GPA gt3.0 ( name , AT_SPSU ) Jorge, Jill,
Wang ClassifGrad (name, AT_SPSU) Jorge,
Wang)
7
More interesting Example of Relation (cont.)

• Putting AT_SPSU relation in a tabular form as
  follows
• AT_SPSU
  Table

name
classif.
age
GPA
grad
Jorge
23
3.2
19
soph
Jill
3.7
Chaudary
18
fresh
2.7
21
sen
Mike
2.8
Look familiar to some of you ?
Wang
22
3.4
grad
Select name from AT_SPSU where GPA gt 3.0 Select
name from AT_SPSU where classif Grad
8
Simple Operators on Relations

• Domain operator, dom, applied on a binary
  relation provides the left elements of the
  ordered pairs.
• Example
• Users jim, tom, sally, joe, kim
• Files f1, f2, f3, f4, f7, f9, f23
• File_owners is defined over Users x Files
• File_owners (jim, f2), (jim, f4), (joe, f9)
• dom ( File_owners) jim, Joe

What does (dom(File_owners) Joe) files,
File_Owners mean to you?
9
Simple Operators on Relations

• Range operator, ran, applied on a binary relation
  provides the right elements of the ordered
  pairs.
• Example
• Users jim, tom, sally, joe, kim
• Files f1, f2, f3, f4, f7, f9, f23
• File_owners is defined over Users x Files
• File_owners (jim, f2), (jim, f4), (joe, f9)
• ran (File_owners) f2, f4, f9

10
Inverse Operator

• The inverse operator applied to a relation, R,
  reverses the relation.
• If R is a relation, then R-1 is the inverse of R.
• Formally (a,b) R ? (b,a) R-1
• Example
• let relation, R (-6, 12), (-7, 14), (-12,
  24)
• Then R-1 (12, -6), (14, -7), (24, -12)

Note that dom R ran R-1
11
some English statements relations

• High-Priority jobs are those with priority levels
  of 5 or above.
• Let the set of jobs be J
• Let the set of priority levels be P
• The cross product is J x P, and
  High-Priority-jobs is a relation over J x P
• High-Priority-jobs a J, b P I bgt5 (a,
  b)
• Customers who purchased more than 1,000
• Let the set of customer be C
• Let the set of dollars spent be Spent
• The cross product is C x Spent
• dom a C, b Spent I b gt1000 (a, b)

Are these formal definitions more helpful under
some circumstances? Your thoughts?