# Is = (equality) a partial order relation?

We know that a partial order relation is a relation which is reflexive , antisymmetric and transitive. Example: (x,y) belongs to R iff x=y. For A={1,2,3}, we get R= {(1,1), (2,2), (3,3)}. Now R is reflexive, transitive and also anti-symmetric (if xRy and yRx then x=y).

So equality should be a partial order relation. Is it so? If yes, then why many authors don't mention it as an example of partial order relation. I only find <= , >= , divides, integral multiple and inclusion as an example in most of the books. I am confused.

• Yes, it is. I suspect that the main reason that it’s often not explicitly mentioned as an example of a partial order is that it’s quite atypical, since in general partial orders are not symmetric. Also, it will certainly be given as an example of an equivalence relation — it is the prototypical one, after all — and some authors may fear that exhibiting it as a partial order as well will confuse many students unless the author carefully explains that it is both, which takes space that the author might prefer to spend on something else. And of course an author might want to leave it as an ... – Brian M. Scott Sep 25 '13 at 6:43
• ... exercise to prove that equality on any set is a partial order on that set. – Brian M. Scott Sep 25 '13 at 6:43
• Equality is the smallest partial order in some sense. – copper.hat Sep 25 '13 at 6:53
• Thank you @brian. You cleared my doubt. – user221458 Sep 25 '13 at 6:53
• @user221458: You’re welcome. – Brian M. Scott Sep 25 '13 at 6:55