Set is Subset of Union

Theorem

The union of two sets is a superset of each:

$S \subseteq S \cup T$

$T \subseteq S \cup T$

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.

Then:

$\ds \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

Let $\mathbb S$ be a set of sets.

Then:

$\ds \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

In the context of a family of sets, the result can be presented as follows:

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Then:

$\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$

where $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\family {S_\alpha}$.

$\ds x \in S$

$\leadsto$

$\ds x \in S \lor x \in T$

$\ds $

$\leadsto$

$\ds x \in S \cup T$

Definition of Set Union

$\ds $

$\leadsto$

$\ds S \subseteq S \cup T$

Definition of Subset

Similarly for $T$.

$\blacksquare$

1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.4$
1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 8 \ \text{(f)}$
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.4$. Union: Example $15$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 6$
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7$: Unions and Intersections
1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.1 \ \text{(ii)}$