How are joins Commutative and Associative? I have read all over the place that joins are associative and commutative.

So A join (B join C) should be the same as (A join C) join B.

But I have a really hard time understanding how this can be so. Assume that A has a property in common with B and B has a property in common with C, but A and C share no common properties to join on.

It seems that in that case, in the second instance (A join C) join B, when the operation took place to join A and C it would result in an empty set, making the join with B impossible.

Whereas in the first case it would work because B join C would result in the expected joined table that would have a property to join with A.
You've made an incorrect assumption.

If three's a relationship between A and B, and between B and C but no relationship between A and C, joining A and C doesn't give you no rows. It actually gives you m * n rows, where m is the number of rows in A, and n is the number of rows in B.

A JOIN B on (a.id = b.A_id)
  JOIN C on (b.id = c.B_id)


becomes, if we join A and C first:

A JOIN C on (1 = 1)
  JOIN B on (a.id = b.A_id AND b.id = c.B_id)


NOTE: College was over 30 years ago - hopefully you understand conceptually, and can rewrite into the correct syntax for actual relational algebra.

If you have 20 rows in A, 30 in B, and 40 in C, and every row in B matches one and only one row in A, and every row in C matches one and only one row in B. When you Join A to B, you'll get 30 rows back; when you join those 30 rows to C, you'll get 40 rows back.

If you first join A to C, with no relationship to define which pairs of rows are valid, you'll get back every possible pair of rows - 800 rows. When we join B to the 800 rows, the connection between B and A means that a row in B can only match the AC rows that include one specific A row - there'll be 40 of those (one for each C row). Out of those 40 the only actual matches are the ones where the connection between B and C is also valid. And, we know that the C rows that match the current B row cannot match any other B row. So, for one B row we might have 2 AC rows that match, for another 4, for yet another just 1.

That said, we know every C row does match one (and only one) B row, so the total number of matches will come out to 40 again - the same 40 matches as if we had first matches A and B, or B and C.

So - the condition on a join doesn't tell you which rows do match - it really tells you which rows don't match. Take it away, and you've got a CROSS JOIN - the cross product of the rows in the two tables.