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inverse image
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(Definition)
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Let $f: A \longrightarrow B$ be a function, and let $U \subset B$ be a subset. The inverse image of $U$ is the set $f^{-1}(U) \subset A$ consisting of all elements $a \in A$ such that $f(a) \in U$
The inverse image commutes with all set operations: For any collection $\{U_i\}_{i \in I}$ of subsets of $B$ we have the following identities for
- Unions: $$f^{-1}\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i \in I} f^{-1}(U_i)$$
- Intersections: $$f^{-1}\left(\bigcap_{i \in I} U_i\right) = \bigcap_{i \in I} f^{-1}(U_i)$$
and for any subsets $U$ and $V$ of $B$ we have identities for
- Complements: $$\left(f^{-1}(U)\right)^\complement = f^{-1}(U^\complement)$$
- Set differences: $$f^{-1}(U \setminus V) = f^{-1}(U) \setminus f^{-1}(V)$$
- Symmetric differences: $$f^{-1}(U \bigtriangleup V) = f^{-1}(U) \bigtriangleup f^{-1}(V)$$
In addition, for $X \subset A$ and $Y \subset B$ the inverse image satisfies the miscellaneous identities
- $(f|_X)^{-1}(Y)=X\cap f^{-1}(Y)$
- $f\left(f^{-1}(Y)\right) = Y\cap f(A)$
- $X \subset f^{-1}(f(X))$ with equality if $f$ is injective.
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"inverse image" is owned by djao. [ full author list (2) | owner history (1) ]
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Cross-references: injective, equality, satisfies, symmetric differences, set differences, complements, intersections, unions, identities, collection, operations, subset, function
There are 45 references to this entry.
This is version 5 of inverse image, born on 2001-10-21, modified 2003-07-30.
Object id is 442, canonical name is InverseImage.
Accessed 15757 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) |
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Pending Errata and Addenda
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