Minimum and Maximum Number of Tuples Consider Relations R and S where are R is having m tuples and S is having n tuples . m&lt;=n . What would be the minimum and maximum number of tuples in each of the following cases (Assume that nothing is mentioned about key constraints)


R union S
R intersection S
R-S
S-R
R NATURAL JOIN S
R Left OUTER Join S 
R/S


My Work


R UNION S


max : n+m ( union we add all the tuples from both relations)

min: 0 ( taking m=n=null )


R INTERSECTION S


max : m ( m &lt; n both relation contains same keys then we may get maximum m keys )

min: 0 ( taking m=n=null if no common keys in both relations)


R - S


max : m ( if they are disjoint then in R-S we will get all tuples of R )

min: 0 ( if all tuples in R is also present in S) 


S-R


max : n ( as explained above )

min: 0 ( as explained above )


R natural join S


max : n*m ( if no matching key constraints natural join will produce Cartesian product )

min: m ( m &lt; n  when key constraints are taken into consideration )


R LEFT OUTER JOIN S


max : m ( everything from left table will be output even if no match)

min: 0 ( when m=0 )


R/S


max : m ( when n=0 )

min: Im Not able to make a  conclusion
Yes, your answers are mostly right, except a few mistakes:

1. R UNION S


max : n+m ( union we add all the tuples from both relations)
Correct, when R and S have no common tuple.
min: 0 ( taking m=n=null )
Wrong, the minimum is n (the greatest of the two sizes, m and n). When all the tuples of R also exist in S.
And m and n cannot be null, they are the sizes of the relations, they are numbers (integers).


2. R INTERSECTION S


max : m ( m&lt;n both relation contains same keys then we may get maximum m keys )
Correct but the reasoning is wrong. You compare tuples of the two relations, not keys.
min: 0 ( taking m=n=null if no common keys in both relations)
Correct but the reasoning is wrong. The result can be 0 because the two relations may have no common tuples).


3. R - S


max : m ( if they are disjoint then in R-S we will get all tuples of R )
Correct
min: 0 ( if all tuples in R is also present in S)
Correct


4. S - R


max : n ( as explained above )
Correct
min: 0 ( as explained above )
Wrong, the minimum is n - m.


5. R natural join S


max : n*m ( if no matching key constraints natural join will produce Cartesian product )
Correct
min: m ( m &lt; n  when key constraints are taken into consideration )
Wrong, the minimum is 0. You can easily find an example, identical with case 2 (INTERSECTION).


6. R LEFT OUTER JOIN S


max : m ( everything from left table will be output even if no match)
Wrong, the maximum is m * n, the same as for natural join. Or just take ON TRUE.
min: 0 ( when m=0 )
Wrong, the minimum is m. Example can be the same as for NATURAL join above (or just take ON FALSE) but it cannot give as a result lees than the number of tuples in R (the left relation in the join).


7. R / S


max : m ( when n=0 )
Correct but it doesn't have to be n=0 or m=0. You can find another example.
min: I'm not able to make a conclusion.
Minimum is 0 Consider that relational division is similar to integer division. 3 / 7 gives 0 in integer division for example. Try to convert this into relational division.