1. What Are the Associative Laws?
The associative laws explain that when taking the union or intersection of three sets, the way the sets are grouped does not change the final result. Only the sets involved matter, not the placement of brackets.
This means you can simplify longer expressions without worrying about the grouping order.
2. Associative Law for Union
This law states that when combining three sets using union, the grouping does not affect the outcome.
2.1. Formula
\( (A \cup B) \cup C = A \cup (B \cup C) \)
2.2. Example
If A = {1}, B = {2}, and C = {3}, then both ways give:
\( (A \cup B) \cup C = \{1,2,3\} \)
\( A \cup (B \cup C) = \{1,2,3\} \)
3. Associative Law for Intersection
The same idea applies to intersection: the grouping of sets does not change the intersection result.
3.1. Formula
\( (A \cap B) \cap C = A \cap (B \cap C) \)
3.2. Example
If A = {1,2,3}, B = {2,3}, and C = {3}, then both expressions give:
\( (A \cap B) \cap C = \{3\} \)
\( A \cap (B \cap C) = \{3\} \)
4. Why Associative Laws Matter
Associative laws make it easier to simplify long set expressions and help avoid mistakes when working with multiple sets. They allow union and intersection to be handled in any convenient grouping.
5. Important Points
Key ideas to remember:
5.1. Key Ideas
- Grouping does not affect the union or intersection.
- Brackets can be rearranged freely in expressions with only ∪ or only ∩.
- The laws apply to any sets A, B, and C.