Preimage of Intersection under Mapping/Family of Sets

Theorem

Let $S$ and $T$ be sets.

Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.

Let $f: S \to T$ be a mapping.

Then:

$\ds f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$

where:

$\ds \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.

$f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.

As $f$ is a mapping, it is by definition also a many-to-one relation.

It follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.

$\ds \RR \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} \RR \sqbrk {T_i}$

where $\RR \sqbrk {T_i}$ denotes the image of $T_i$ under $\RR$.

$\blacksquare$

$\ds x$

$\in$

$\ds f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i}$

$\ds \leadstoandfrom \ \ $

$\ds \map f x$

$\in$

$\ds \bigcap_{i \mathop \in I} T_i$

$\ds \leadstoandfrom \ \ $

$\ds \forall i \in I: \, $

$\ds \map f x$

$\in$

$\ds T_i$

$\ds \leadstoandfrom \ \ $

$\ds \forall i \in I: \, $

$\ds x$

$\in$

$\ds f^{-1} \sqbrk {T_i}$

$\ds \leadstoandfrom \ \ $

$\ds x$

$\in$

$\ds \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$

$\blacksquare$

1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages: Remark $2$
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.12$: Set Inclusions for Image and Inverse Image Sets: Theorem $12.6 \ \text{(c)}$
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $1$
1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions: Exercise $4 \ \text{(b})$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions: Exercise $1.6.3 \ \text{(ii)}$
1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions: Problem $\text{A}.3.1$
2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 2$