Consider this Scenario :
R (A,B) : 10000 rows ( showing Seperate Values Of A and B for a tuple/row)
A Tuples ( a1,a2,.......,a1001, (a1001,........ a1001) 8999 times ) B Tuples ( b1,b2,b3,.........,b10000 )
S (A,C) : 5000 rows ( showing Seperate Values Of A and C for a tuple/row)
A Tuples ( a1,a2,.......,a1001, (a1001,........ a1001) 3999 times ) C Tuples ( c1,c2,c3,.........,c5000 )
Now Making Join Of Both (Natural Join) ( For each tuple of R matching with Each tuple of S on basis of Common Column A)
For a1 : Match Found : 1 ; a2 : Match Found 1 ; ................ ; a1000 : Match Found 1 ; (Inidividual Match (A) :1000 )
For a1001 :
Match Found : For R -> A ( row no 1001) : With S -> A (row no 1001)
For R -> A ( row no 1001) : With S -> A (row no 1002)
... So 4000 matches for R -> A ( row no 1001 ) .... Similarly 4000 matches for R -> ( row no 1002) ....
For Others : 8999 a1001's of relation R -> Matches by Duplicate (B) : 9000 * 4000 = 36 * 10^6
Total Matches : (A) + (B) : 36000000 + 1000 = 36001000. (Answer)
P.S. : Here we are taking as many as duplicates in Relation R and S ; because we need to find maximum number of tuples.