An equivalence relation is a binary relation that fulfills reflexivity, symmetry and transitivity.A partially ordered set is a set equipped with a binary relation that fulfills reflexivity, antisymmetry and transitivity.For a partially ordered set, some elements of the set might be incomparable, i.e. x ≤ y or y ≤ x may both not hold. That's why we use the term"partially". An example would be ⊂ as a binary relation and the set M:={{1},{2},{1,2}}. Obviously, {1} and {2} are incomparable.Question: Can also elements of a set, equipped with an equivalence relation, be incomparable?

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An equivalence relation is a binary relation that fulfills reflexivity, symmetry and transitivity.A partially ordered set is a set equipped with a binary relation that fulfills reflexivity, antisymmetry and transitivity.For a partially ordered set, some elements of the set might be incomparable, i.e.   x  ≤  y or   y  ≤  x may both not hold. That&#039;s why we use the term&quot;partially&quot;. An example would be   ⊂ as a binary relation and the set M:={{1},{2},{1,2}}. Obviously, {1} and {2} are incomparable.Question: Can also elements of a set, equipped with an equivalence relation, be incomparable?

sleuteleni7 · Accepted Answer

The word &quot;incomparable&quot; in the statement of your question is a bit misleading. Technically speaking, the question does not make sense because for an equivalence relation the notion of &quot;comparable&quot; is not defined. Of course, you may wish to say that if two elements of the set are related by the given equivalence relation, then they are &quot;comparable&quot;, but that does not do justice to the notion of equivalence relation. Moreover, if you choose to do so, then the answer to the question is in general &#039;yes&#039;, because if you have more than one equivalence class you already have &quot;incomparable&quot; (i.e. non-related) elements.It is best to think about an equivalence relation as a tool to identify elements of a set, whereas an order relation, partial or linear, is a tool to create a hierarchy. Identification and Hierarchy are two essentially different notions.

An equivalence relation is a binary relation that fulfills reflexivity, symmetry and transitivity.

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