) – Thus n ( P ) = Hence teachers teach physics. Example In a class of students, like to play cricket and like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football ?
Solution Let X be the set of students who like to play cricket and Y be the set of students who like to play football. Then X ∪ Y is the set of students who like to play at least one game, and X ∩ Y is the set of students who like to play both games. Given n ( X) = , n ( Y ) = , n ( X ∪ Y ) = , n (X ∩ Y) = ? Using the formula n ( X ∪ Y ) = n ( X ) + n ( Y ) – n ( X ∩ Y ), we get = + – n (X ∩ Y) Fig .
SETS Thus, n (X ∩ Y) = i.e., students like to play both games. Example In a survey of students in a school, were listed as taking apple juice, as taking orange juice and were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice. Solution Let U denote the set of surveyed students and A denote the set of students taking apple juice and B denote the set of students taking orange juice.
Then n (U) = , n (A) = , n (B) = and n (A ∩ B) = . Now n (A ′ ∩ B ′ ) = n (A ∪ B) ′ = n (U) – n (A ∪ B) = n (U) – n (A) – n (B) + n (A ∩ B) = – – + = Hence students were taking neither apple juice nor