Editing Commutative law on sets (section)

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

The '''commutative law''' is an important property of some set operations, meaning the order in which we perform the operation does not affect the result.

== Commutative Law for Union ==

For any two sets <math>A</math> and <math>B</math>, the union operation is commutative. This means:

<math>
A \cup B = B \cup A
</math>

In words, combining set <math>A</math> with set <math>B</math> is the same as combining set <math>B</math> with set <math>A</math>.

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

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

Both are the same set (order does not matter in sets), so  
<math>A \cup B = B \cup A</math>.

== Commutative Law for Intersection ==

Similarly, the intersection operation is also commutative. For any two sets <math>A</math> and <math>B</math>:

<math>
A \cap B = B \cap A
</math>

This means the set of common elements between <math>A</math> and <math>B</math> is the same regardless of the order.

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

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

So,  
<math>A \cap B = B \cap A</math>.

== Summary ==

* The commutative law holds for both union and intersection of sets.  
* Changing the order of the sets does not change the result.  
* This property helps simplify calculations and reasoning in set theory.

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