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≳ 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 $(x',y')\succsim (x,y)$]
It can be easily shown that the indifference relation derived from the reflexive, transitive and complete weak preference relation in the way described above will be an equivalence relation. It is important because this relation partitions the entire commodity space into indifference classes, where two consumption vectors belong to a class if and only if the individual is indifferent between the two. These classes are also known as indifference curves. This further helps in solving related problems.
