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
- $P = Q$
- 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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $11$