Mapping Image of Union

Theorem

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

Let $A$ and $B$ be subsets of $S$.

Then:

$f \left({A \cup B}\right) = f \left({A}\right) \cup f \left({B}\right)$

General Theorem

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

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

$\displaystyle f \left({\bigcup \mathbb S}\right) = \bigcup_{X \in \mathbb S} f \left({X}\right)$

Proof

As $f$, being a mapping, is also a relation, we can apply Image of Union:

$\mathcal R \left({A \cup B}\right) = \mathcal R \left({A}\right) \cup \mathcal R \left({B}\right)$

and

$\displaystyle \mathcal R \left({\bigcup \mathbb S}\right) = \bigcup_{X \in \mathbb S} \mathcal R \left({X}\right)$

$\blacksquare$

Also see

• Mapping Preimage of Union
• Mapping Image of Intersection
• Mapping Preimage of Intersection

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 10$: Inverses and Composites
• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $12.13 \ \text{(a)}$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.3$: Theorem $3$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 12 \alpha$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 5$: Theorem $5.1 \ \text{(ii)}$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$: Exercise $1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 21.4 \ \text{(ii)}$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.3$: Problem $\text{A}.3.1$