Discrete Mathematics Relations

1
Discrete MathematicsRelations
2
What is a relation

• Relation generalizes the notion of functions.
• Recall A function takes EACH element from a set
  and maps it to a UNIQUE element in another set
• f X ? Y
• ? x ? X, ? y such that f(x) y
• Let A and B be sets.
• A binary relation R from A to B is a subset of A
  ? B
• Recall A x B (a, b) a ? A, b ? B
• aRb (a, b) ? R.
• Application
• Relational database model is based on the concept
  of relation.

3
What is a relation

• Example
• Let A be the students in a the CS major
• A Ayse, Baris, Canan, Davut
• Let B be the courses the department offers
• B BIM111, BIM122, BIM124
• We specify relation R ? A ? B as the set that
  lists all students a ? A enrolled in class b ? B
• R (Ayse, BIM111), (Baris, BIM122), (Baris,
  BIM124), (Davut, BIM122), (Davut, BIM124)

4
More relation examples

• Another relation example
• Let A be the cities in Turkey
• Let B be the districts in Turkey
• We define R to mean a is a district in city b
• Thus, the following are in our relation
• (Bakirköy, Istanbul)
• (Keçiören, Ankara)
• (Nilüfer, Bursa)
• (Tepebasi, Eskisehir)
• etc
• Most relations we will see deal with ordered
  pairs of integers

5
Representing relations
We can represent relations graphically
We can represent relations in a table
6
Relations vs. functions

• If R ? X ? Y is a relation, then is R a function?
• If f X ? Y is a function, then is f a relation?

7
Relations on a set

• A relation on the set A is a relation from A to A
• In other words, the domain and co-domain are the
  same set
• We will generally be studying relations of this
  type

8
Relations on a set

• Let A be the set 1, 2, 3, 4
• Which ordered pairs are in the relation
• R (a,b) a divides b
• R (1,1), (1,2), (1,3), (1,4), (2,2), (2,4),
  (3,3), (4,4)

9
More examples

• Consider some relations on the set Z
• Are the following ordered pairs in the relation?
• (1,1) (1,2) (2,1) (1,-1) (2,2)
• R1 (a,b) ab
• R2 (a,b) agtb
• R3 (a,b) ab
• R4 (a,b) ab
• R5 (a,b) ab1
• R6 (a,b) ab3

X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
10
Relation properties

• Six properties of relations we will study
• Reflexive
• Irreflexive
• Symmetric
• Asymmetric
• Antisymmetric
• Transitive

11
Reflexivity vs. Irreflexivity

• Reflexivity
• Definition A relation is reflexive if
• (a,a) ? R for all a ? A
• Irreflexivity
• Definition A relation is irreflexive if
• (a,a) ? R for all a ? A
• Examples
• Is the divides relation on Z reflexive?
• Is the ? (not ?) relation on a P(A) irreflexive?

?
?
gt
lt

o
o
x
x
o
reflexive
irreflexive
x
x
o
o
x
12
Reflexivity vs. Irreflexivity

• A relation can be neither reflexive nor
  irreflexive
• Example?
• A 1, 2, R (1, 1)
• It is not reflexive, since (2, 2) ? R,
• It is not irreflexive, since (1, 1) ? R.

13
Symmetry, Asymmetry, Antisymmetry

• A relation is symmetric if
• for all a, b ? A, (a,b) ? R? (b,a) ? R
• A relation is asymmetric if
• for all a, b ? A, (a,b) ? R ? (b,a) ? R
• A relation is antisymmetric if
• for all a, b ? A, ((a,b) ? R ? (b,a) ? R) ? ab
• (Second definition) for all a, b ? A, ((a,b) ? R
  ? a ? b) ? (b,a) ? R)

isTwinOf
?
?

gt
lt
o
x
x
o
x
x
symmetric
x
x
x
x
o
o
asymmetric
x
o
o
o
o
o
antisymmetric
14
Notes on symmetric relations

• A relation can be neither symmetric or asymmetric
• R (a,b) ab
• This is not symmetric
• -4 is not related to itself
• This is not asymmetric
• 4 is related to itself
• Note that it is antisymmetric

15
Transitivity

• A relation is transitive if
• for all a, b, c ? A, ((a,b)?R ? (b,c)?R) ?
  (a,c)?R
• If a lt b and b lt c, then a lt c
• Thus, lt is transitive
• If a b and b c, then a c
• Thus, is transitive

16
Transitivity examples

• Consider isAncestorOf()
• Let Ayse be Bariss ancestor, and Baris be
  Canans ancestor
• Thus, Ayse is an ancestor of Baris, and Baris is
  an ancestor of Canan
• Thus, Ayse is an ancestor of Canan
• Thus, isAncestorOf() is a transitive relation
• Consider isParentOf()
• Let Ayse be Bariss parent, and Baris be Canans
  parent
• Thus, Ayse is a parent of Baris, and Baris is a
  parent of Canan
• However, Ayse is not a parent of Canan
• Thus, isParentOf() is not a transitive relation

17
Summary of properties of relations
() Alternative definition
18
Combining relations

• There are two ways to combine relations R1 and R2
• Via Set operators
• Via relation composition

19
Combining relations via Set operators

• Consider two relations R and R
• R U R all numbers OR
• Thats all the numbers
• R n R all numbers AND
• Thats all numbers equal to
• R ? R all numbers or , but not both
• Thats all numbers not equal to
• R - R all numbers that are not also
• Thats all numbers strictly greater than
• R - R all numbers that are not also
• Thats all numbers strictly less than
• Note that its possible the result is the empty
  set

20
Combining via relational composition

• Similar to function composition
• Let R be a relation from A to B, and S be a
  relation from B to C
• Let a ? A, b ? B, and c ? C
• Let (a,b) ? R, and (b,c) ? S
• Then the composite of R and S consists of the
  ordered pairs (a,c)
• We denote the relation by S ? R
• Note that S comes first when writing the
  composition!
• (a, c) ? S ? R if ? b such that (a, b) ? R, and
  (b,c) ? S

21
Combining via relational composition

• Let M be the relation is mother of
• Let F be the relation is father of
• What is M ? F?
• If (a,b) ? F, then a is the father of b
• If (b,c) ? M, then b is the mother of c
• Thus, M ? F denotes the relation maternal
  grandfather
• What is F ? M?
• If (a,b) ? M, then a is the mother of b
• If (b,c) ? F, then b is the father of c
• Thus, F ? M denotes the relation paternal
  grandmother
• What is M ? M?
• If (a,b) ? M, then a is the mother of b
• If (b,c) ? M, then b is the mother of c
• Thus, M ? M denotes the relation maternal
  grandmother

22
Combining via relational composition

• Given relation R
• R ? R can be denoted by R2
• R2 ? R (R ? R) ? R R3
• Example M3 is your mothers mothers mother