View source for 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 <math>A</math>, <math>B</math>, and <math>C</math>:

<math>
A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
</math>

This means the intersection of <math>A</math> with the union of <math>B</math> and <math>C</math> is equal to the union of the intersections of <math>A</math> with <math>B</math> and <math>A</math> with <math>C</math>.

=== Example 1 ===  
Let  
<math>A = \{1, 2, 3, 4\}</math>  
<math>B = \{3, 4, 5\}</math>  
<math>C = \{4, 5, 6\}</math>  

Step 1: Calculate <math>B \cup C</math>:  
<math>B \cup C = \{3, 4, 5, 6\}</math>  

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

Step 3: Calculate <math>A \cap B</math> and <math>A \cap C</math>:  
<math>A \cap B = \{3, 4\}</math>  
<math>A \cap C = \{4\}</math>  

Step 4: Calculate <math>(A \cap B) \cup (A \cap C)</math>:  
<math>\{3, 4\} \cup \{4\} = \{3, 4\}</math>  

Both sides are equal:  
<math>A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = \{3, 4\}</math>

== Distributive Law of Union over Intersection ==

Similarly, union distributes over intersection:

<math>
A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
</math>

This means the union of <math>A</math> with the intersection of <math>B</math> and <math>C</math> is equal to the intersection of the unions of <math>A</math> with <math>B</math> and <math>A</math> with <math>C</math>.

=== Example 2 ===  
Let  
<math>A = \{1, 2, 3\}</math>  
<math>B = \{2, 3, 4\}</math>  
<math>C = \{3, 4, 5\}</math>  

Step 1: Calculate <math>B \cap C</math>:  
<math>B \cap C = \{3, 4\}</math>  

Step 2: Calculate <math>A \cup (B \cap C)</math>:  
<math>\{1, 2, 3\} \cup \{3, 4\} = \{1, 2, 3, 4\}</math>  

Step 3: Calculate <math>A \cup B</math> and <math>A \cup C</math>:  
<math>A \cup B = \{1, 2, 3, 4\}</math>  
<math>A \cup C = \{1, 2, 3, 4, 5\}</math>  

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

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

== Summary ==

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

[[Category:Set Theory]]
[[Category:Set Operations]]
[[Category:Mathematics]]