Given bags R and S where a tuple t appears n and m times respectively, the intersection
of bags R and S will have tuple t appear min( n, m ) times. The further intersection of bag
T with the tuple t appearing o times will produce tuple t min( o, min( n, m ) ) times in the
final result.
On the right-hand side:
Given bags S and T where a tuple t appears m and o times respectively, the intersection of
bags R and S will have tuple t appear min( m, o ) times. The further intersection of bag R
with the tuple t appearing n times will produce tuple t min( n, min( m, o ) ) times in the
final result.
The intersection of bags R,S and T will yield a result where tuple t appears min( n,m,o ) times.
For sets:
This is a similar case when dealing with bags except the tuple t can only appear at most once in
each set. The tuple t only appears in the result if all the sets have the tuple t. Otherwise, the tuple
t will not appear in the result.
For bags:
On the left-hand side:
Given that tuple r in R, which appears m times, can successfully join with tuple s in S,
which appears n times, we expect the result to contain mn copies. Also given that tuple t
in T, which appears o times, can successfully join with the joined tuples of r and s, we
expect the final result to have mno copies.
On the right-hand side:
Given that tuple s in S, which appears n times, can successfully join with tuple t in T,
which appears o times, we expect the result to contain no copies. Also given that tuple r
in R, which appears m times, can successfully join with the joined tuples of s and t, we
expect the final result to have nom copies.
The order in which we perform the natural join does not matter for bags.
For sets:
This is a similar case when dealing with bags except the joined tuples can only appear at most
once in each result. If there are tuples r,s,t in relations R,S,T that can successfully join, then the
result will contain a tuple with the schema of their joined attributes.
For bags:
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