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 $A$ , $B$ , and $C$ :

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

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

Example: Union

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

Calculate $(A \cup B) \cup C$ :

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

Calculate $A \cup (B \cup C)$ :

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

Both are equal:

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

Associative Law for Intersection

Similarly, the intersection operation is associative:

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

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

Example: Intersection

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

Calculate $(A \cap B) \cap C$ :

$A \cap B = {2, 3}$ $(A \cap B) \cap C = {2, 3} \cap {3, 4, 5} = {3}$

Calculate $A \cap (B \cap C)$ :

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

Both are equal:

$(A \cap B) \cap C = A \cap (B \cap C) = {3}$

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.