Prove the following 1 Given a poset S we define a relation

Prove the following

(1) Given a poset (S, ), we define a relation on S by x y if and only if x y and x not equal to y. The relation is a pseudo-order.

In order to prove the poset as pseudo order, we need to prove two things

1) The given poset relation must be irreflexive

2) The given relation must be transitive

In order to prove it is irreflexive, we must prove that (x,x) must not belong to the relation

relation is x < y and x not equal to y

Hence (x,x) can never belong to relation on S

Hence it is irreflexive in nature

Let the relation is satisfied (x,y) and (y,z) then we need to prove it is also satisfied for (x,z)

Since x <= y and x not equal to y ------- (i)

Since y <= z and y not equal to z ------- (i)

adding the equations (i) and (ii)

x <= z and since it is greater than x hence it satisfies the relation

Hence the given relation is both irreflexive and transitive

Hence the given relation is a pseudo order relation