1. What Are the Commutative Laws?
The commutative laws state that the order of sets does not affect the result of union or intersection. Switching A and B gives the same outcome.
This makes calculations simpler, because you don't have to worry about which set comes first.
2. Commutative Law for Union
The union of two sets remains the same even if the order is changed.
2.1. Formula
\( A \cup B = B \cup A \)
2.2. Example
If A = {1,2} and B = {2,3}, then:
\( A \cup B = \{1,2,3\} = B \cup A \)
3. Commutative Law for Intersection
The intersection of two sets also stays the same when the order is reversed.
3.1. Formula
\( A \cap B = B \cap A \)
3.2. Example
If A = {a,b,c} and B = {b,c,d}, then:
\( A \cap B = \{b,c\} = B \cap A \)
4. Why Commutative Laws Matter
The laws help simplify set expressions and make operations easier to work with. They save time and reduce confusion, especially in longer problems.
5. Important Points
A few quick notes about commutative laws:
5.1. Key Ideas
- Order does not affect union or intersection.
- Useful for rearranging expressions.
- These laws hold for any sets A and B.