of car owners investigated, M be the set of persons who owned car A and S be the set of persons who owned car B. Given that n ( U ) = , n (M ) = , n ( S ) = and n ( S ∩ M ) = . Then n ( S ∪ M ) = n ( S ) + n ( M ) – n ( S ∩ M ) = + – = But S ∪ M ⊂ U implies n ( S ∪ M ) ≤ n ( U ). This is a contradiction.
So, the given data is incorrect. Example A college awarded medals in football, in basketball and in cricket. If these medals went to a total of men and only three men got medals in all the three sports, how many received medals in exactly two of the three sports ? MATHEMATICS Solution Let F, B and C denote the set of men who received medals in football, basketball and cricket, respectively.
Then n ( F ) = , n ( B ) = , n ( C ) = n (F ∪ B ∪ C ) = and n (F ∩ B ∩ C ) = Therefore, n (F ∪ B ∪ C ) = n ( F ) + n ( B ) + n ( C ) – n (F ∩ B ) – n (F ∩ C ) – n (B ∩ C ) + n ( F ∩ B ∩ C ), gives n ( F ∩ B ) + n ( F ∩ C ) + n ( B ∩ C ) = Consider the Venn diagram as given in Fig . Here, a denotes the number of men who got medals in football and basketball only, b denotes the number of men who got medals in football and cricket only, c denotes the number of men who got medals in basket ball and cricket only and d