View source for Associative Law of Sets

= Associative Law of Sets - Definition, Explanation, and Examples =

The '''associative law''' is a fundamental property of set operations which states that when performing the same operation multiple times, the grouping (or association) of sets does not affect the result.

== Associative Law for Union ==

For any three sets <math>A</math>, <math>B</math>, and <math>C</math>:

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

This means that whether you first unite <math>A</math> and <math>B</math> and then unite the result with <math>C</math>, or first unite <math>B</math> and <math>C</math> and then unite the result with <math>A</math>, the final set is the same.

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

Calculate <math>(A \cup B) \cup C</math>:

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

Calculate <math>A \cup (B \cup C)</math>:

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

Both are equal:

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

== Associative Law for Intersection ==

Similarly, the intersection operation is associative:

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

This means that whether you first find the intersection of <math>A</math> and <math>B</math>, and then intersect with <math>C</math>, or first find the intersection of <math>B</math> and <math>C</math>, and then intersect with <math>A</math>, the final set is the same.

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

Calculate <math>(A \cap B) \cap C</math>:

<math>A \cap B = \{2, 3\}</math>  
<math>(A \cap B) \cap C = \{2, 3\} \cap \{3, 4, 5\} = \{3\}</math>

Calculate <math>A \cap (B \cap C)</math>:

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

Both are equal:

<math>(A \cap B) \cap C = A \cap (B \cap C) = \{3\}</math>

== Summary ==

* The associative law allows us to group sets in any way when performing unions or intersections without changing the result.  
* It simplifies complex expressions by removing the need to worry about parentheses.

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