Commutative law on sets

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 $A$ and $B$, the union operation is commutative. This means:

$A\cup B=B\cup A$

In words, combining set $A$ with set $B$ is the same as combining set $B$ with set $A$.

Example: Union

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

Then: $A\cup B=\{1,2,3,4,5\}$ and $B\cup A=\{3,4,5,1,2\}$

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

Commutative Law for Intersection

Similarly, the intersection operation is also commutative. For any two sets $A$ and $B$:

$A\cap B=B\cap A$

This means the set of common elements between $A$ and $B$ is the same regardless of the order.

Example: Intersection

Let $A=\{1,2,3,4\}$ $B=\{3,4,5,6\}$

Then: $A\cap B=\{3,4\}$ and $B\cap A=\{3,4\}$

So, $A\cap B=B\cap A$.

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.