# Preference ordering relation

≳ is a preference ordering if it is reflexive , transitive and complete. In Mathematics relations are said to be in a equivalence relation if they are reflexive, symmetric and transitive.

Can we call preference ordering a equivalence relation? And if so what is the significance of preference relation being a equivalence relation in further studies of microeconomics.?

No reflexivity, transitivity and completeness does not imply symmetry. For example: Consider the preference relation $\succsim$ over $\mathbb{R}^2_+$ defined as:
$(x,y)\succsim (x',y')$ if and only if $xy \geq x'y'$
Such preferences have a utility representation $u(x, y) = xy$, and therefore they are reflexive, transitive and complete. However, these are not symmetric because $(1,2)\succsim (1,1)$ but $(1,1)\not\succsim (1,2)$.
But we can define the indifference relation $\sim$ from the weak preference relation $\succsim$ in this way:
$(x,y)\sim (x',y')$ if and only if [$(x,y)\succsim (x',y')$ and $(x',y')\succsim (x,y)$]