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Collection Inverse Image Union Intersection Multivalued Function Set Function Set Difference Complement Property Equality Satisfaction Relation

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Direct Image

Let $f\colon A \longrightarrow B$ be a function, and let $U \subset A$ be a subset. The direct image of is the set $f(U) \subset B$ consisting of all elements of which equal for some $u \in U$ .

Direct images satisfy the following properties:

Unions: For any collection $\{U_i\}_{i \in I}$ of subsets of ,

$\displaystyle f\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i\in I} f(U_i).$
Intersections: For any collection $\{U_i\}_{i \in I}$ of subsets of ,

$\displaystyle f\left(\bigcap_{i \in I} U_i\right) \subset \bigcap_{i\in I} f(U_i).$
Set difference: For any $U,V \subset A$ ,

$\displaystyle f(V \setminus U) \supset f(V) \setminus f(U).$
In particular, the complement of satisfies $f(U^\complement) \supset f(A) \setminus f(U)$ .
Subsets: If $U \subset V \subset A$ , then $f(U) \subset f(V) \subset B$ .
Inverse image of a direct image: For any $U \subset A$ ,

$\displaystyle f^{-1}(f(U)) \supset U$
with equality if is injective.
Direct image of an inverse image: For any $V \subset B$ ,

$\displaystyle f(f^{-1}(V)) \subset V$
with equality if is surjective.

1	Equality
1	Property
1	Collection
1	Union
1	Complement
1	Function
1	Set
1	Inverse Image
1	Set Difference
1	Multivalued Function
1	Intersection
2	Satisfaction Relation