∈ P ( A ) ∩ P ( B). This gives P ( A ∩ B ) ⊂ P ( A ) ∩ P ( B ). Let Y ∈ P ( A ) ∩ P ( B ). Then Y ∈ P ( A) and Y ∈ P ( B ).
So, Y ⊂ A and Y ⊂ B. Therefore, Y ⊂ A ∩ B, which implies Y ∈ P ( A ∩ B ). This gives P ( A ) ∩ P ( B ) ⊂ P ( A ∩ B) Hence P ( A ∩ B ) = P ( A ) ∩ P ( B ). Example A market research group conducted a survey of consumers and reported that consumers like product A and consumers like product B, what is the least number that must have liked both products?
Solution Let U be the set of consumers questioned, S be the set of consumers who liked the product A and T be the set of consumers who like the product B. Given that n ( U ) = , n ( S ) = , n ( T ) = So n ( S ∪ T ) = n ( S ) + n ( T ) – n ( S ∩ T ) = + – n (S ∩ T) = – n ( S ∩ T ) Therefore, n ( S ∪ T ) is maximum when n ( S ∩ T ) is least. But S ∪ T ⊂ U implies n ( S ∪ T ) ≤ n ( U ) = . So, maximum values of n ( S ∪ T ) is .
Thus, the least value of n ( S ∩ T ) is . Hence, the least number of consumers who liked both products is . Example Out of car owners investigated, owned car A and owned car B, owned both A and B cars. Is this data correct?
Solution Let U be the set