Absorption Laws (Set Theory)/Union Absorbs Intersection

Theorem

$S \cup \paren {S \cap T} = S$

$\ds $

$\ds \paren {S \cap T} \subseteq S$

$\ds $

$\leadsto$

$\ds S \cup \paren {S \cap T} = S$

$\blacksquare$

$\ds x$

$\in$

$\ds S \cup \paren {S \cap T}$

$\ds \leadstoandfrom \ \ $

$\ds x$

$\in$

$\ds S \lor \paren {x \in S \land x \in T}$

Definition of Set Intersection and Definition of Set Union

$\ds \leadstoandfrom \ \ $

$\ds x$

$\in$

$\ds S$

$\blacksquare$

Intersection Absorbs Union, where it is proved that $S \cap \paren {S \cup T} = S$

These two results together are known as the Absorption Laws, corresponding to the equivalent results in logic.

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $3$
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $3$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): absorption laws
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): absorption laws
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): absorption laws
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): absorption laws