CS 103 Discrete Structures

Lecture 19

Relations

RelationsChapter 9

Chapter SummaryRelations and Their Properties

Relations and Their Properties

Section 9.1

Section SummaryRelations and FunctionsProperties of Relations

Reflexive RelationsSymmetric and Antisymmetric RelationsTransitive Relations

Combining Relations

Binary RelationsA relation is a subset of the Cartesian product

Relations can be used to solve problems such as:Determining which pairs of cities are linked by

airline flights in a networkFinding a feasible order for the different

phases of a complicated projectProducing a useful way to store information in

computer databases

A binary relation R from a set A to a set B is a subset R ⊆ A × B. Therefore, R consists of ordered pairs where the 1st element of each ordered pair comes from A and the 2nd element comes from B

Binary Relations(a, b) R means a is related to b by R

a R b denotes (a, b) R

a R b denotes (a, b) R

Example 1: Let A be the set of students, B be the set of courses and R be the relation that consists of pairs(a, b), where a is a student enrolled in course b. Then,If Ahmad and Ali are enrolled in CS103, then

(Ahmad, CS103) R and (Ali, CS103) RIf Ahmad is also enrolled in CS111, then

(Ahmad, CS111) RIf Ali is not enrolled in CS111, then (Ali, CS111)

R

Binary RelationsExample 2: Let A be the set of cities, B be the set of regions. (a, b) belongs to R if city a is in region b

Then (Al-Mahd, Al-Madinah), (Gada, Makkah), and(Al-Zolfy, Ryiadh) are in R

Example 3A = {0, 1, 2} B = {a, b}

R = {(0, a), (0, b), (1, a), (2, b)} is a relation from A to B

We also canrepresent the relation from theset A to the set Bgraphically or using a table

Binary Relations on a Set A binary relation on the set A is a relation from A to A. In other words, a relation on a set A is a subset of A A

Example 1: A = {a, b, c}. Then R = {(a, a),(a, b), (a, c)} is a relation on A

Example 2: A = {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)}

How Many Relations on a Set?Because a relation on A is the same thing as a subset of A × A, we count the subsets of A × A

As A × A has |A|2 elements, and a set with m

elements has 2m subsets, therefore A × A has 2|A|2 subsets

there are 2|A|2 relations on a set A

Binary Relations on a Set: Example 3Which of these relations on the set of integers contain each of the pairs (1, 1), (1, 2), (2, 1), (1, -1), and (2, 2)?

R1 = {(a, b) | a ≤ b}R2 = {(a, b) | a > b}R3 = {(a, b) | a = b or a = - b}R4 = {(a, b) | a = b}R5 = {(a, b) | a = b + 1}R6 = {(a, b) | a + b ≤ 3}

(1, 1)is in R1, R3, R4, and R6

(1, 2)is in R1 and R6

(2, 1) is in R2, R5, and R6

(1, -1) is in R2, R3, and R6

(2, 2)is in R1, R3, and R4

Note that these relations are on an infinite set and each of these relations is an infinite set

Properties of Relations: Reflexive

Properties are used to classify relations on a set

A relation R on a set A is called reflexive iff ∀x[x ∊ A ⟶ (x, x) ∊ R], i.e. (a, a) R for every elementa A

A= {1, 2, 3, 4}R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1)}R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4),

(3, 3), (3, 4), (4, 4)}R6 = {(3, 4)}

Properties of Relations: Irreflexive

Relation R on a set A is irreflexive if (a, a) R for all a A

Some relations are neither reflexive nor irreflexive

Example

Let A = {1, 2} and R = {(1, 1)}It is not reflexive, because (2, 2) RIt is not irreflexive, because (1, 1) R

Properties of Relations: SymmetricRelation R on a set A is symmetric iff (b, a) R whenever (a, b) R, for all a, b ARelation R on a set A is antisymmetric iff (a, b) R & (b, a) R then a = b, for all a, b A

Relations can be symmetric & antisymmetric, simultaneously

A = {1, 2, 3, 4}R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1)}R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3),

(3, 4), (4, 4)}R6 = {(3, 4)}

Antisymmetric Property: Example DetailsR1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}

Whenever 2nd-last column shows Yes, last column does not show Yes. R5 is not Antisymmetric

(a, b)

(b, a)

(b, a) R4?

a=b?

(1, 1)

(1, 1)

Yes Yes

(1, 2)

(2, 1)

Yes No

(2, 1)

(1, 2)

Yes No

(2, 2)

(2, 2)

Yes Yes

(3, 4)

(4, 3)

No -

(4, 1)

(1, 4)

No -

(4, 4)

(4, 4)

Yes Yes

Antisymmetric Property: Example Details

R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}

Whenever 2nd-last column shows Yes, last column also shows Yes. R5 is Antisymmetric

(a, b)

(b, a)

(b, a) R4?

a=b?

(2, 1)

(1, 2)

No -

(3, 1)

(1, 3)

No -

(3, 2)

(2, 3)

No -

(4, 1)

(1, 4)

No -

(4, 2)

(2, 4)

No -

(4, 3)

(3, 4)

No -

Antisymmetric Property: Example DetailsR5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}

Whenever 2nd-last column shows Yes, last column also shows Yes. R5 is Antisymmetric

(a, b)

(b, a)

(b, a) R4?

a=b?

(1, 1)

(1, 1)

Yes Yes

(1, 2)

(2, 1)

No -

(1, 3)

(3, 1)

No -

(1, 4)

(4, 1)

No -

(2, 2)

(2, 2)

Yes Yes

(2, 3)

(3, 2)

No -

(2, 4)

(4, 2)

No -

(3, 3)

(3, 3)

Yes Yes

(3, 4)

(4, 3)

No -

(4, 4)

(4, 4)

Yes Yes

Properties of Relations: TransitiveRelation R on a set A is transitive

if whenever (a, b) R and (b, c) R, then (a, c) R,for all a, b, c A

A = {1, 2, 3, 4}R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1)}

R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}

R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}R6 = {(3, 4)}

Transitive Property: Example DetailsR1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}

Start with the first pair, (1,1). Here a=1, b=1.Are their pairs that have their 1st elements as b? Yes

Only one pair, (1, 2). Here b=1, c=2Is (a, c) a member of R1? Yes. R1 can be transitive.

Now move to the next pair, (1, 2). Here a=1, b=2.Are their pairs that have their 1st elements as b? Yes

Two pairs, (2, 1), (2, 2). Here b=2, c=1 and c=2Is (a, c) a member of R1? Yes for both c=1 and c=2. R1 may be transitive.

Now move to the next pair, (2, 1). Here a=2, b=1.Are their pairs that have their 1st elements as b? Yes

Two pairs, (1, 1), (1, 2). Here b=1, c=1 and c=2Is (a, c) a member of R1? Yes for both c=1 and c=2. R1 may be transitive.

Now move to the next pair, (2, 2). Here a=2, b=2.Are their pairs that have their 1st elements as b? Yes

Two pairs, (2, 1), (2, 2). Here b=2, c=1 and c=2Is (a, c) a member of R1? Yes for both c=1 and c=2. R1 may be transitive.

Now move to the next pair, (3, 4). Here a=3, b=4.Are their pairs that have their 1st elements as b? Yes

Two pairs, (4, 1), (4, 4). Here b=4, c=1 and c=4Is (a, c) a member of R1? No for c=1. R1 is not transitive.

Finished! No need to do any further checking.

Transitive Property: Example DetailsR4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}

There are no “No” answers in the last column. Therefore, the relation R4 is transitive.

(a, b)

Pairs with b as their 1st element

c (a, c)R4?

(2, 1)

- - -

(3, 1)

- - -

(3, 2)

(2, 1) 1 Yes

(4, 1)

- - -

(4, 2)

(2, 1) 1 Yes

(4, 3)

(3, 1)(3, 2)

12

YesYes

Properties of Relations: Summary

Reflexive aA, (a, a)R

Irreflexive aA, (a, a)R

Symmetric a,bA, (a, b)R (b, a)R

Antisymmetric

a,bA, (a, b)R (b, a)R a = b

Transitive a,b,cA, (a, b)R (b, c)R (a, c)R

Combining RelationsRelations are sets

They can be combined in any way sets can be combined

Example 1

Let A = {1, 2, 3}, B = {1, 2, 3, 4},R1 = {(1, 1), (2, 2), (3, 3)}

R2 = {(1, 1), (1, 2), (1, 3), (1, 4)}

Combining Relations: Example 2R1 = {(x, y) | x < y} & R2 = {(x, y) | x > y}, where x,yR

R1 R2 = ?(x, y) R1 R2 iff (x, y) R1 or (x, y) R2

(x, y) R1 R2 iff x < y or x > yx < y or x > y implies that x yR1 R2 = {(x, y) | x y}

R1 R2 = as (x, y) cannot belong to both R1 & R2

R1 - R2 = R1

R2 - R1 = R2

R1 R2 = R1 R2 - R1 R2 = {(x, y) | x y}

x > y

x < y

y- axi

s

x-axis

x = y

R is a relation from set A to set BS is a relation from set B to set C

The composite of R and S, S R, is the relation consisting of ordered pairs (a, c), where a A,c C, and for which there exists an elementb B such that (a, b) R and (b, c) S

ExampleR is the relation from A = {1, 2, 3} to B = {1, 2, 3, 4}

R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} S is the relation from B = {1, 2, 3, 4} to C = {0, 1, 2}

S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}S R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}

b

c

a

a b c

Composite of Relations

Powers of a RelationFor relation R on set A, powers R

n, n = 1,

2, 3, ... , are defined recursively by

Example

R = {(1, 1), (2, 1), (3, 2), (4, 3)} = R1

R2 = R R = {(1, 1), (2, 1), (3, 1), (4, 2)}

R3 = R

2 R = {(1, 1), (2, 1), (3, 1), (4, 1)}

R4 = R

3 R = {(1, 1), (2, 1), (3, 1), (4, 1)} = R

3

Rn = R

3 for n = 5, 6, 7, .... as well

Section 9.1: Exercises

Section 9.1: Exercises

Section 9.1: Exercises

Section 9.1: Exercises

Section 9.1: Exercises