Absorption Laws (Set Theory)/Union with Intersection

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof 1

\(\displaystyle \) \(\) \(\displaystyle \paren {S \cap T} \subseteq S\) Intersection is Subset
\(\displaystyle \) \(\leadsto\) \(\displaystyle S \cup \paren {S \cap T} = S\) Union with Superset is Superset‎

$\blacksquare$


Proof 2

\(\displaystyle x\) \(\in\) \(\displaystyle S \cup \paren {S \cap T}\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S \lor \paren {x \in S \land x \in T}\) Definition of Set Intersection and Definition of Set Union
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S\) Disjunction Absorbs Conjunction

$\blacksquare$


Also see

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


Sources