πa,b(πa,b(r)×s−πR(r))

= (r/s)

2. Expanding logically the statement means to select t (a,b) from r such that for all tuples u in s, there is a tuple v in r, such that u = v[S] and t = v[R-S]. This is just equivalent to

(r/s)

3. Expanding logically the statement means that select t (a,b) from r such that for all tuples v in r, there is a tuple u in s, such that u = v[S] and t = v[R-S]. This is equivalent to saying to select (a,b) values from r, where the c value is in (s/r), which will be true only if c in r is a foreign key referring to c in s.

4. This selects (a,b) from all tuples from r which has an equivalent c value in s.

So, 1 and 2 are equivalent