Proof that Inner Join is Commutative
An inner join is the subset of rows from the cartesian product where a certain condition is true. Although the cartesian product is not commutative (nor associative), it is with regard to relational theory. 

It is so because the order of attributes doesn't matter in relational theory. If A and B are tables, then:

A x B = { (a1, a2, .., an, b1, b2, .. bn) | (a1..an) € A and (b1..bn) € B}
      = { (b1, b2, .., bn, a1, a2, .. an) | (a1..an) € A and (b1..bn) € B}  ((1))
      = { (b1, b2, .., bn, a1, a2, .. an) | (b1..bn) € B and (a1..an) € A}  ((2))
      = B x A                                                               ((3))


((1)) because the order of attributes doesn't matter
((2)) because the logical and is commutative
((3)) by definition of B x A  

The condition for choosing which elements of the cartesian must be selected will be the same no matter the JOIN order, and as such, doesn't influence this reasoning.

The proof for associativity follows the same line of reasoning.