Minimum and Maximum number of Tuples in the Following Relations

Question

Given R with n tuples S with m tuples n<m  then How many minimum and maximum tuples in follwing relations . Please Justify with Reason / Examples

• R-S
• S-R
• R Left Join S
• R  Natural Join S
• R/S
• S/R

Comments

Confused ! What is correct one ? :(

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@pc,if no tuple is common then we go for cartesian product.am i right??

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Exactly what I have told above. Even shared few References :)

Answer 1

[Someone should verify . This is what I think]

1. R UNION S

• max : n+m
  Reason : union we add all the tuples from both relations. ie When R and S have no common tuple.

• min: n
  Reason : The minimum is n (the greatest of the two sizes, m and n). When all the tuples of R also exist in S.

2. R INTERSECTION S

• max : m ( m<n )
  Reason : both relation contains same tuples then we may get maximum m keys

• min: 0
  Reason : taking m=n=null if no common tuples in both relations

3. R - S

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

• min: 0
  Reason : if all tuples in R is also present in S
   

4. S - R

• max : n 
  Reason : as explained above
   

• min: n-m
  Reason : m<n  there will be some tuples in S after deleting the common tuples

5. R natural join S

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

• min:  0
  Reason : Identical with case 2 (INTERSECTION).

6. R LEFT OUTER JOIN S

• max : m*n
  Reason : if all rows in left tables matches with all rows in right table

• min: m  (could be 0 when m= 0)
  Reason : The minimum is 1 when m=1 , minimum is 2 when m=2, minimum is 0 when m=0

7. R / S

• max : m 
  Reason :   when n=0

• min: 0
  Reason :  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

Comments

For 7th point, if m=0 then R / S  has zero tuples. Isn't it?

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6.max of left natural join

How will all rows of left table match with all of right?

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For natural join

mn is max.

Case 1: if there is a common attribute between and , and every row of R matches with the each row of S - i.e., the join attribute has the same value in all the rows of both and ,
Case 2: If there is no common attribute between R and S .

Min is 0.

If There is a common attribute between R and S and nothing matches.