Distributive Laws/Set Theory/Examples/A cap B cap (C cup D) subset of (A cap D) cup (B cap C)

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Example of Use of Distributive Laws of Set Theory

Let:

$P = A \cap B \cap \paren {C \cup D}$
$Q = \paren {A \cap D} \cup \paren {B \cap C}$

Then:

$P \subseteq Q$


Corollary

$P = Q$

if and only if:

both $B \cap C \subseteq A$ and $A \cap D \subseteq B$


Proof

\(\ds P\) \(=\) \(\ds A \cap B \cap \paren {C \cup D}\)
\(\ds \) \(=\) \(\ds \paren {A \cap B \cap C} \cup \paren {A \cap B \cap D}\) Intersection Distributes over Union


Let $P$ be expressed as:

$P = X \cup Y$

where:

$X = A \cap B \cap D$
$Y = A \cap B \cap C$


Then:

\(\ds X\) \(=\) \(\ds A \cap B \cap D\)
\(\ds \) \(\subseteq\) \(\ds A \cap D\) Intersection is Subset
\(\ds \) \(\subseteq\) \(\ds Q\) Set is Subset of Union

and:

\(\ds Y\) \(=\) \(\ds A \cap B \cap C\)
\(\ds \) \(\subseteq\) \(\ds B \cap C\) Intersection is Subset
\(\ds \) \(\subseteq\) \(\ds Q\) Set is Subset of Union

and so:

\(\ds X \cup Y\) \(\subseteq\) \(\ds Q\) Union is Smallest Superset
\(\ds \leadsto \ \ \) \(\ds P\) \(\subseteq\) \(\ds Q\)

$\blacksquare$


Sources

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