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

Corollary to $A \cap B \cap \paren {C \cup D} \subseteq \paren {A \cap D} \cup \paren {B \cap C}$
Let:
 * $P = A \cap B \cap \paren {C \cup D}$
 * $Q = \paren {A \cap D} \cup \paren {B \cap C}$

Then:
 * $P = Q$


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

Sufficient Condition
Let $P = Q$.

and:

Thus we have both:
 * $B \cap C \subseteq A$

and
 * $A \cap D \subseteq B$

Necessary Condition
Let $B \cap C \subseteq A$ and $A \cap D \subseteq B$.

We have:

and:

and so:

and so $P = Q$ as required.