Distributive Law of Sets

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.