inverse image (original) (raw)

The inverse image commutes with all set operations: For any collection {Ui}i∈I of subsets of B, we have the following identities for

1. 1. Unions:

      f-1⁢(⋃i∈IUi)=⋃i∈If-1⁢(Ui)

and for any subsets U and V of B, we have identities for

In addition, for X⊂A and Y⊂B, the inverse image satisfies the miscellaneous identities

1. 1. (f|X)-1⁢(Y)=X∩f-1⁢(Y)
2. 1. f⁢(f-1⁢(Y))=Y∩f⁢(A)
3. 1. X⊂f-1⁢(f⁢(X)), with equality if f is injective.