The necessity of a reflexive relation

Portuga

Gentlemen: I was wondering about equivalence, reflexive, transitive and symmetric relations, and I realized the importance of the transitive and symmetric ones, but it was not so intuitive for me to make explicit the reflexive relation... can someone explain why it is necessary to make explicit this relation when considering equivalence?
Sorry for my poor english and thanks in advance.

Number Nine

There are a number of reasons; the most obvious is that reflexivity is not always true, and so you have to specify when it is.

Relations in general were developed in order to rigorously abstract away from certain common, well behaved relationship (e.g. equality, less than or equal to, etc.). When you do that, you immediately run into the fact that the kinds of relations that are most interesting (equivalence relations and partial orderings, say) have the property that any element is related to itself. This is not true in general; for instance, take the "less than" relation defined on the reals; it is not true that x < x (for any x). When it's true (as in the case of an equivalence relation), you specify that it's true. When it's not, you don't.

Portuga

Ok, mister number nine, I understood. Thank you.

Stephen Tashi

There is a need to state the reflexive property explicity since not all useful relations are reflexive - for example, the relation "<". We wouldn't want people assuming that 3 < 3.

Portuga

Ok, mr. Stephen, thank you!