Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 263

Then • the reflexive closure of Ris the smallest by inclusion reflexive binary relationRrefon Xsuch thatR ⊆ Rref; • the symmetric closure of Ris the smallest by inclusion symmetric binar

k

Applications: Mathematical Proofs and Automated Reasoning 239

5.2.5 Special binary relations

Definition 201 A binary relation R ⊆ X2is called:

• reflexive if it satisfies ∀ x(xRx );

• irreflexive if it satisfies ∀ x ¬( xRx ), that is, if X2− R is reflexive;

• serial if it satisfies ∀ x ∃ y(xRy );

• functional if it satisfies ∀ x ∃! y(xRy ), where ∃! y means “there exists a unique y ;”

• symmetric if it satisfies ∀ x ∀ y(xRy → yRx );

• asymmetric if it satisfies ∀ x ∀ y(xRy → ¬ yRx );

• antisymmetric if it satisfies ∀ x ∀ y(xRy ∧ yRx → x=y );

• connected if it satisfies ∀ x ∀ y(xRy ∨ yRx ∨ x=y );

• transitive if it satisfies ∀ x ∀ y ∀ z((xRy ∧ yRz)→ xRz );

• an equivalence relation if it is reflexive, symmetric, and transitive;

• euclidean if it satisfies ∀ x ∀ y ∀ z((xRy ∧ xRz)→ yRz );

• a pre-order (or quasi-order) if it is reflexive and transitive;

• a partial order if it is reflexive, transitive, and antisymmetric, that is, an antisymmetric

pre-order;

• a strict partial order if it is irreflexive and transitive;

• a linear order (or total order) if it is a connected partial order; or

• a strict linear order (or strict total order) if it is a connected strict partial order.

Proposition 202 For any set X and binary relation R ⊆ X2:

(a) R is reflexive iff E X ⊆ R ;

(b) R is symmetric iff R −1 ⊆ R iff R −1 =R ;

(c) R is asymmetric iff R −1 ∩ R=∅;

(d) R is antisymmetric iff R −1 ∩ R ⊆ E X ;

(e) R is connected iff R ∪ R −1 ∪ E X =X2;

(f) R is transitive iff R2⊆ R

Functions can be regarded as special type of relations by means of their graphs: the

graphof a functionf :A → Bcan be defined as the binary relationG f ⊆ A × Bwhere

G f ={( a, f(a))| a ∈ A } A relation R ⊆ A × B is therefore functional iff it is the graph of a function fromAtoB(exercise)

LetR ⊆ X × Xbe a binary relation on a setX Then

• the reflexive closure of Ris the smallest by inclusion reflexive binary relationRrefon

Xsuch thatR ⊆ Rref;

• the symmetric closure of Ris the smallest by inclusion symmetric binary relationRsym

onXsuch thatR ⊆ Rsym; and

• the transitive closure of Ris the smallest by inclusion transitive binary relationRtran

onXsuch thatR ⊆ Rtran

As an exercise, show that each of these closures always exists