reﬂexivity, symmetry, and transitivitypeople.hsc.edu/faculty-staff/robbk/math262/lectures/spring...

Reflexivity, Symmetry, and TransitivityLecture 38Section 8.2

Robb T. Koether

Hampden-Sydney College

Mon, Apr 1, 2013

Robb T. Koether (Hampden-Sydney College) Reflexivity, Symmetry, and Transitivity Mon, Apr 1, 2013 1 / 23

1 Three Properties

2 Closures

3 Assignment


Outline

1 Three Properties

2 Closures

3 Assignment


Three Properties

Definition (Reflexive, Symmetric, Transitive)Let R be a relation on a set A. Then

R is reflexive if, for all x ∈ A, (x , x) ∈ R.R is symmetric if, for all x , y ∈ A,

(x , y) ∈ R → (y , x) ∈ R.

R is transitive if, for all x , y , z ∈ A,

((x , y) ∈ R and (y , z) ∈ R)→ (x , z) ∈ R.


Three Properties

Reflexivity: Every element of the set A has the relation to itself.Symmetry: If any element x ∈ A has the relation to some elementy ∈ A, then y has the same relation to x .Transitivity: If any element x ∈ A has the relation to some elementy ∈ A and that element y has the same relation to some elementz ∈ A, then x has that relation to z.


Examples

Define a relation R on Z by (x , y) ∈ R if 5 | (x − y).Show that R is reflexive, symmetric, and transitive.


Examples

Which of the following relations are reflexive? symmetric?transitive?

a | b on Z.gcd(a, b) > 1 on Z.x × y < 0 on R.A ⊆ B on P(X ).p → q on a set of statements.p ∧ q ≡ p on a set of statements.


Outline

1 Three Properties

2 Closures

3 Assignment


The Reflexive Closure

Definition (Reflexive closure)Let A be a set and let R be a relation on A. The reflexive closure of R,denoted Rr , is

R ∪ {(a, a) | a ∈ A}.

Rr is the smallest relation on A that contains R and is reflexive.


The Graph of the Reflexive Closure

b

a

d f

c

e

b

a

d f

c

e

Add loops to get the graph of the reflexive closure.


Examples

What is the reflexive closure of < on R?What is the reflexive closure of ⊂ on P(X )?What is the reflexive closure of ≡ modn on Z.


The Symmetric Closure

Definition (Symmetric closure)Let A be a set and let R be a relation on A. The symmetric closure ofR, denoted Rs, is

R ∪ {(b, a) | (a, b) ∈ R}.

Rs is the smallest relation on A that contains R and is symmetric.


The Graph of the Symmetric Closure

b

a

d f

c

e

b

a

d f

c

e

Make every arrow double-ended to get the graph of the symmetricclosure.


Examples

What is the symmetric closure of < on R?What is the symmetric closure of ⊂ on P(X )?What is the symmetric closure of ≡ modn on Z.


The Transitive Closure

Definition (Transitive closure)Let A be a set and let R be a relation on A. The transitive closure of R,denoted Rt , is the smallest subset of A× A that contains R and istransitive.

To find the transitive closure of a relation requires an algorithm.It is not enough to define

Rt = R ∪ {(a, c) | (a, b), (b, c) ∈ R}.

Why not?


The Transitive Closure

Given a relation R on a set A, the following procedure will produceRt .

Let n = 1 and let R0 = ∅ and R1 = R.While Rn 6= Rn−1, do the following.

Let Rn+1 = Rn ∪ {(a, c) | (a, b), (b, c) ∈ Rn}.n← n + 1.


The Graph of the Transitive Closure

b

a

d f

c

e

b

a

d f

c

e

R1 R2

Connect all back-to-back arrows.


The Graph of the Transitive Closure

b

a

d f

c

e

b

a

d f

c

e

R2 R3

Connect all back-to-back arrows again.


Examples

What is the transitive closure of (a, b) ∈ R on Z defined byb = a + 1?What is the transitive closure of (A, B) ∈ R on P({1, 2, 3}) definedby A ∩ B 6= ∅?What is the transitive closure of (A, B) ∈ R on P(Z) defined by|A− B| <∞.


All Three Closures

b

a

d f

c

e

b

a

d f

c

e

We can do all three closures at the same time.


Example

Define R on Z by (a, b) ∈ R if a− b = 2.Find the reflexive, symmetric, and transitive closure of R.


Outline

1 Three Properties

2 Closures

3 Assignment


Assignment

AssignmentRead Sections 8.2, pages 449 - 457.Exercises 2, 10, 14, 17, 19, 22, 23, 29, 35, 36, 52, 53, page 458.


Collected Homework 10

Collected Homework 10Exercises 11, 17, page 426.Exercises 9, 12, 15, page 439.Exercises 11, 14, page 448.


reﬂexivity, symmetry, and transitivitypeople.hsc.edu/faculty-staff/robbk/math262/lectures/spring...

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