binary relations

what is a binary relation

A set is a binary relation if and only if it contains only ordered

pairs

A binary relation R is reflexive on a set S

iff for all elements d of S the pair ⟨d,d⟩ is an element

of R

A binary relation R is symmetric on a Set S

iff for all elements d,e of S: if ⟨d,e⟩ e R then

⟨e,d⟩ e R.

A binary relation R is asymmetric on a set S

iff for no elements d,e of S: ⟨d,e⟩ e R and ⟨e,d⟩ e

R

A binary relation R is antisymmetric on a set S

iff for no two distinct (that is, different) elements

d,e of S: ⟨d,e⟩ e R and ⟨e,d⟩ e R

A binary relation R is transitive on a set S

iff for all elements d,e,f of S: if ⟨d,e⟩ e R and

⟨e,f⟩ e R, then also ⟨d,f⟩ e R

A binary relation R is an equivalence relation on S

iff R is reflexive on S, symmetric on S, and transitive on S

A binary relation Ris a function

iff for all d,e,f: if ⟨d,e⟩ e R and ⟨d,f⟩ e R then e= f

The domain of a function R is…

the set {d: There is an esuch that ⟨d,e⟩ e R}

The range of a function R is…

the set {e: There is a d such that ⟨d,e⟩ e R}

R is a function into the set M…

iff all elements of the range of the function are in M

If d is in the domain of a function R one writes…

R(d) for the unique object e such that ⟨d,e⟩is in R