Distributive Law of Sets - Definition, Explanation, and Examples

The distributive law shows how union and intersection operations distribute over each other. It is a key property in set theory that helps simplify expressions involving both operations.

Distributive Law of Intersection over Union

For any three sets $A$ , $B$ , and $C$ :

$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

This means the intersection of $A$ with the union of $B$ and $C$ is equal to the union of the intersections of $A$ with $B$ and $A$ with $C$ .

Example 1

Let $A = {1, 2, 3, 4}$ $B = {3, 4, 5}$ $C = {4, 5, 6}$

Step 1: Calculate $B \cup C$ : $B \cup C = {3, 4, 5, 6}$

Step 2: Calculate $A \cap (B \cup C)$ : $A \cap {3, 4, 5, 6} = {3, 4}$

Step 3: Calculate $A \cap B$ and $A \cap C$ : $A \cap B = {3, 4}$ $A \cap C = {4}$

Step 4: Calculate $(A \cap B) \cup (A \cap C)$ : ${3, 4} \cup {4} = {3, 4}$

Both sides are equal: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = {3, 4}$

Distributive Law of Union over Intersection

Similarly, union distributes over intersection:

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

This means the union of $A$ with the intersection of $B$ and $C$ is equal to the intersection of the unions of $A$ with $B$ and $A$ with $C$ .

Example 2

Let $A = {1, 2, 3}$ $B = {2, 3, 4}$ $C = {3, 4, 5}$

Step 1: Calculate $B \cap C$ : $B \cap C = {3, 4}$

Step 2: Calculate $A \cup (B \cap C)$ : ${1, 2, 3} \cup {3, 4} = {1, 2, 3, 4}$

Step 3: Calculate $A \cup B$ and $A \cup C$ : $A \cup B = {1, 2, 3, 4}$ $A \cup C = {1, 2, 3, 4, 5}$

Step 4: Calculate $(A \cup B) \cap (A \cup C)$ : ${1, 2, 3, 4} \cap {1, 2, 3, 4, 5} = {1, 2, 3, 4}$

Both sides are equal: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C) = {1, 2, 3, 4}$

Summary

The distributive laws help simplify expressions involving both union and intersection.
Intersection distributes over union, and union distributes over intersection.