(1) proves that reflexivity, antisymmetry and transitivity are satisfied.

Reflexivity: Because the relation R is partially ordered, it is reflexive, xRx? xSx

So s also satisfies reflexivity.

Anti-symmetry: Because the relation R is partial order and satisfies anti-symmetry, when x≠y

xSy？ yRx xRy ySx

Therefore, S also satisfies antisymmetry.

Transitivity:

Because the relation r is partial order and satisfies transitivity, then

xSy ∧ ySz？ yRx ∧ zRy？ zRx？ xSz

So s also satisfies transitivity.

(2)

R is less than or equal to

Then s is greater than or equal to the relation.