Talk:Equivalence of Definitions of Saturated Set Under Equivalence Relation

An alternative proof of 2 implies 1 would go as follows:

Let $T = \ds \bigcup_{u \mathop \in U} \eqclass u {}$ with $U \subset S$.

By Image of Union under Mapping, $\map q T = \ds \bigcup_{u \mathop \in U} \map q {\eqclass u {} }$.

By Preimage of Union under Mapping, $\map {q^{-1} } {\map q T} = \ds \bigcup_{u \mathop \in U} \map {q^{-1} } {\map q {\eqclass u {} } }$.

It then requires something like the fact that equivalence classes are saturated. This could be proved here (which would clutter the page somewhat) or on a separate page. Disadvantage of the latter is that we're, strictly speaking, not allowed to talk about saturations when we're still defining it, so in order to not risk circular reasonings, I decided to put the argument here for who's interested. (Or in case someone wants to write out the details and add its proof here.) Either way, it is a consequence of the 2nd definition that an equivalence class is saturated. --barto (talk) 14:58, 16 March 2017 (EDT)