Let A be a set Show that there is a natural one to one corre

Let A be a set. Show that there is a natural one to one correspondence between partial order relations on A and strict partial order relations on A.

Solution

Let we have two relations E,F such that <A,E> is called a strict partial ordered set and <A,F> is called the partial ordered set. A one to one correspondence between strict partial orderings E and F on a set is given by F = E D and E = F\\D, where D is the diagonal D:={<x,x>:xA} and further explaination will be explained by the given 2 proofs.

a) For any x,y,z A

Since it is given that < is the relation on A defined by x < y therefore we can say,

Not x < x,

Not y < y and

Not z < z (Irreflexivity Property)

If x < y then not y < x this implies that x y (Asymmetry Property)

If x < y and y < z then x < z. (Transitivity Property).

Hence < is called a strict partial order relation because it satisfies all three properties of strict partial order relation.

So we can say, if is a non-partial order relation then the corresponding strict partial order < is the irreflexive closure given by x < y if x y or x y

b) For any x,y,z A

Since it is given that is the relation on A defined by x y therefore we can say,

x x, y y and z z (Reflexivity Property)

If x y and y x this implies that x = y (Antisymmetry Property - Atmost one relation between 2 distinct elements)

If x y and y z then x z. (Transitivity Property).

Hence is called a partial order relation because it satisfies all three properties of partial order relation.

So we can say, if < is a strict partial order relation then the corresponding non-partial order is the reflexive closure given by x y if x < y or x = y.

From these 2 proofs we can say that for any relation E and F these conditions hold strict partial relation and non-strict partial relation therefore there is a one to one correspondence between strict partial relation and partial relation on set A.