Relations

1
Relations Functions

• Introductory Discrete Mathematics (CS/MAT165)

2
Cartesian Product

• For sets A and B
• A ? B
• The set of all ordered pairs (a, b) where a ? A
  and b ? B

3
Relation

• For sets A and B
• Binary relation R is a subset of A ? B
• x R y ? (x, y) ? R
• x R y ? (x, y) ? R

4
Relation

• If x R y, then define R

Range
Domain
X
Y
1 2 3
1 3 5
5
Try this!

• Draw a diagram for the following relation
• R (a, 1), (a, 3), (b, 2), (c, 3)

6
And this!

• Define a relationship, R X ? Y, where X is the
  set of all possible numerical grades and Y is set
  of all possible letter grades

7
Function

• A relation F where
• ? x ? A, ? y ? B such that (x, y) ? F, and
• If (x, y) ? F and (x, z) ? F, then y z
• y F(x) ? (x, y) ? F ? F x ? y

8
Inverse Relation

• R-1 (y, x) ? B ? A (x, y) ? R
• Example
• R (1, 2), (1, 3), (2, 3)
• R-1 (2, 1), (3, 1), (3, 2)

9
Definitions

• Binary relationA relationship from set A to
  itself
• R is reflexive ? ? x ? A, x R x
• R is symmetric ? ? x, y ? A, x R y ? y R x
• R is transitive ? ? x, y, z ? A, (x R y) ? (y R
  z) ? x R z

10
Try this

• Given the diagram of relation R, determine
  whether it is reflexive, symmetric, and/or
  transitive

? a
? b
? c
? d
11
Transitive Closure

• Transitive closure of R
• Rt is transitive
• R ? Rt
• If S is a transitive relation, and R ? S,then Rt
  ? S

12
Try this

• Find the transitive closure Rt for the
  followingR (1, 2), (2, 2), (3, 1)
• What is Rt - R ?

13
And this

• Given the set of all real numbers R fill out the
  following table

14
Partition

• For set A, a collection of nonempty, mutually
  disjoint subsets A1 An whose union is A
• The binary relation induced by the partition R is
  defined as? x, y ? Ax R y ? x, y ? Ai

A2
A4
A1
A3
15
Prove!

• If R is a relation induced by a partition on A,
  then R is reflexive, symmetric and transitive

16
Equivalence

• Let A be a set and R a binary relation on A
• R is an equivalence relation iff R is reflexive,
  symmetric, and transitive
• For each element a in A, the equivalence class of
  a, a, also called the class of a, is the set of
  all elements x in A such that x R aa x ? A
  x R a

17
Equivalence

• Let A be a set and R a equivalence relation on A
  and a, b ? A
• (a R b) ? (a b)
• How can we prove this?

18
Equivalence

• Let A be a set and R a equivalence relation on A,
  and a, b ? A
• Either a ? b ? or a b
• How can we prove this?

19
Equivalence

• Let A be a set and R a equivalence relation on A
• The distinct equivalence classes of R form a
  partition of A
• How can we prove this?

20
Partial Order

• Antisymmetric
• ? a, b ? A,(a R b) ? (b R a) ? (a b)
• Let R be a binary relation on set A
• R is a partial order relation iff R is reflexive,
  antisymmetric, and transitive

21
Total Order

• Let R be a partial order relation on set A
• If for any a and b in A, either a R b or b R a,
  then R is a total order relation on A

22
Topological Sorting

• Given partial order relations R and R on set A,
  R is a topological sorting for R iff R is a
  total order compatible with R