A market research group conducted a survey of 1000 consumers and reported that 745 consumers like product A and 430 consumers like product B. What is the least number that must have liked both products ?

Explanation

Let P and Q be the sets of cosumers who like product A and product B respectively.
Then , `n(P) =745 and n( Q) =430` .
Now , `n(Pcup Q) =n(P) +n(Q) -n(Pcap Q)`
`rArr n( Pcup Q) =745 +430 -n(Pcap Q)`.
`rArr n(Pcup Q) =1175 -n(Pcap Q)`.
Clearly, `n(Pcap Q)` is least when ` n(PcupQ)` is maximum and therefore, `n(Pcup Q) =1000`. ltbr. So` 1000=1175 -n(Pcap Q) rArr n(Pcap Q) =(1175-1000)=175`.
Hence, the least number of conumers liking both the products is 175.