Intersection Distributes over Union
Theorem
Set intersection is distributive over set union:
- $R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$
Family of Sets
Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be a indexed family of subsets of a set $S$.
Let $B \subseteq S$.
Then:
- $\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$
where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.
General Result
Let $S$ and $T$ be sets.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T$ be a subset of $\powerset T$.
Then:
- $\ds S \cap \bigcup \mathbb T = \bigcup_{X \mathop \in \mathbb T} \paren {S \cap X}$
Proof
\(\ds \) | \(\) | \(\ds x \in R \cap \paren {S \cup T}\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in R \land \paren {x \in S \lor x \in T}\) | Definition of Set Union and Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in R \land x \in S} \lor \paren {x \in R \land x \in T}\) | Conjunction is Left Distributive over Disjunction | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {R \cap S} \cup \paren {R \cap T}\) | Definition of Set Union and Definition of Set Intersection |
$\blacksquare$
Demonstration by Venn Diagram
In the left hand diagram, $R$ is depicted in yellow and $S \cup T$ is depicted in blue.
Their intersection $R \cap \paren {S \cup T}$ where they overlap is depicted in green.
In the right hand diagram, $\paren {R \cap S}$ is depicted in yellow and $\paren {R \cap T}$ is depicted in blue.
Their intersection, where they overlap, is depicted in green.
Their union is the total shaded area: yellow, blue and green.
As can be seen by inspection, the areas are the same.
Examples
$3$ Arbitrarily Chosen Sets
Let:
\(\ds A\) | \(=\) | \(\ds \set {3, -i, 4, 2 + i, 5}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {-i, 0, -1, 2 + i}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {- \sqrt 2 i, \dfrac 1 2, 3}\) |
Intersection with Union
\(\ds A \cap \paren {B \cup C}\) | \(=\) | \(\ds \set {3, -i, 4, 2 + i, 5} \cap \set {-i, 0, -\sqrt 2 i, -1, 2 + i, \dfrac 1 2, 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {3, -i, 2 + i}\) |
Union of Intersections
\(\ds \paren {A \cap B} \cup \paren {A \cap C}\) | \(=\) | \(\ds \set {-i, 2 + i} \cup \set 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {3, -i, 2 + i}\) |
Thus it is seen that:
- $A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$
Arbitrary Integer Sets
Let:
\(\ds A\) | \(=\) | \(\ds \set {2, 4, 6, 8, \dotsc}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {1, 3, 5, 7, \dotsc}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {1, 2, 3, 4}\) |
Then:
- $\paren {A \cup B} \cap C = \set {1, 2, 3, 4} = \paren {A \cap C} \cup \paren {B \cap C}$
Also known as
The result Intersection Distributes over Union, along with Union Distributes over Intersection, are together known as:
Also see
Sources
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