Both Union and Intersection are commutative.

**Theorem**:

**Proof:**

*Part 1:*

Let . Then or . Therefore, or . Which is to say that . This implies that . And by symmetry, .

Therefore, .

*Part 2*:

Let . Then and . This implies and . So, . Therefore, .

Also, if , then and , which implies that and , and so .

Therefore, .

*Now go lift something heavy,*

Nick Horton