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It's easy to find examples of equivalence relations (for example, A shares room with B), but I can't seem to find a real life example of an order relation (that is, a relation that's reflexive, antisymmetric and transitive).

In particular, I can't seem to find a (real life) relation that is reflexive, yet not symmetric.

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  • $\begingroup$ brothers and sisters. $\endgroup$
    – Felix Marin
    Commented Jul 3, 2014 at 14:50
  • $\begingroup$ @FelixMarin "A is B's brother/sister" is an equivalence relation (if we admit that, by definition, I'm my own brother as I share parents with myself). It is reflexive, symmetric (if A is B's brother/sister, then B is A's brother/sister) and transitive. $\endgroup$
    – Miguelgondu
    Commented Jul 3, 2014 at 17:58
  • $\begingroup$ I was thinking in the age: $\large A "\leq" B \Leftrightarrow {\rm age}\left(A\right) \leq {\rm age}\left(B\right)$. $\endgroup$
    – Felix Marin
    Commented Jul 3, 2014 at 22:26

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Daoud is not richer than Fatma.

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  • $\begingroup$ But is Daoud not richer than himself? $\endgroup$
    – Miguelgondu
    Commented Jul 3, 2014 at 1:45
  • $\begingroup$ Of course! :-) How could one be richer than oneself? $\endgroup$
    – Sergio Parreiras
    Commented Jul 3, 2014 at 1:47
  • $\begingroup$ Oh, My first interpretation was incorrect. You're right. :) $\endgroup$
    – Miguelgondu
    Commented Jul 3, 2014 at 1:50
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    $\begingroup$ And this of course points to a huge family of similar relations: Daoud is not taller than Fatma; Daoud is not older than Fatma; Daoud did not score better than Fatma on the national college entrance examinations, and so forth. $\endgroup$
    – MJD
    Commented Jul 3, 2014 at 3:03
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A relation over geographic regions:

New York State is within the bounds of the United States.

New York City is within the bounds of New York State.

Manhattan is within the bounds of New York City.

Manhattan is within the bounds of Manhattan; where else would it be? Because it is within New York City, it must be within the bounds of New York State, and therefore also within the bounds of the United States.

On the other hand, since New York City and New York State are two different things, not two names for the same thing, the above implies that New York State cannot possibly be within the bounds of New York City.

It is possible for a region to be within the bounds of two other regions, neither of which is within the bounds of the other, but that doesn't violate either the reflexive or transitive property.

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Most simple corporate organizational charts, where every person has at most a single manager, can be seen as an order. Then we say $A\leq B$ if $B$ is some number of steps above $A$, including $0$ steps above.

This example has the advantage of not being "linear-ish."

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  • $\begingroup$ An organizational chart does look like a Hasse Diagram, I had not thought about it! $\endgroup$
    – Miguelgondu
    Commented Jul 3, 2014 at 2:03
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"Taller". I am exactly as tall as myself. If I am taller than Bob and Bob is taller than Mary, then I am taller than Mary. Also, Bob cannot be taller than me.

Alternative: Question numbers at Math StackExchange are totally ordered. This question is #854928.

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An order relation that is used in schools around the world every day: (non-strict) alphabetical order. Define $w_1\preceq w_2$ to mean "either $w_1$ is the same as $w_2$, or $w_1$ comes before $w_2$ in alphabetical order".

Of course we should recognise that it's not altogether easy to give a precise definition of alphabetical order. For example, do capital letters come before or after lowercase? Or do they count as the same? And what about punctuation: does "its" come before "it's"? However these are really linguistic problems rather than mathematical problems, and as long as we can sort out what it actually means, alphabetical order is definitely an example of a partial order.

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