Can a Partial Order be symmetric in addition to its properties?

Question

For example,a relation,R defined on integers. $R = \{(a,b)\mid a^3=b^3\}$

Could this be partial order?

Answer 1

A partial order can be symmetric, but that tells us nothing respect to being a partial order. (Antisymmetry is not the antonym of symmetry. A relation can be both.)

To show that a relation is a partial order:

• You need to show that the relation is reflexive

  • For all $x \in \mathbb Z$, is it true that $x^3 = x^3$?

• You need to show that the relation is antisymmetric

  • For all $x, y \in \mathbb Z$, is it true that IF $\;x^3 = y^3\;$ AND $y^3 = x^3$, then $x = y$?

• You need to show that the relation is transitive:

  • For all $x, y, z \in \mathbb Z$, is it true that IF $x^3 = y^3,\;$ AND $\;y^3 = z^3,\;$ then $x^3 = z^3\;$?

(Note: in this case, the relation is both symmetric and antisymmetric.)