Talk:Preimage of Intersection under Relation

Equality is true
Equality is true if you consider preimages of a map. In particular all of the following are true:

Let $f$ be a map from $A$ to $B$, where $A$ and $B$ are sets. Then the following are true:


 * $f^{-1}(A\cap B)=f^{-1}(A)\cap f^{-1}(B)$


 * $f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B)$


 * $f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B)$

In a while I'll try to see if I can split the page Definition:Preimage into pages about maps and relations. Then I'll add pages for all of these, given they don't exist (I haven't yet checked). JSchoone 17:15, 10 December 2011 (CST)
 * All these are proved in Mapping Preimage of Intersection and its related pages. The condition under which equality holds for $\cap$ and $\setminus$ are also explored in detail (something rarely seen in the literature). --prime mover 17:51, 10 December 2011 (CST)