Chapter 8 - Section 8.5 - Inclusion-Exclusion - Exercises - Page 557: 4

450000

Let $A$ denote the set of computer owners who will buy a printer for their machines next year, and $B$ denote the set of computer owners who will buy at least one software package. It is given that $|A|=650000$, $|B|=1250000$, and $|A\cup B|=1450000$. We are looking for the number of computer owners that will buy both a printer and at least one software package, which is $|A\cap B|$. By the Principle of Inclusion-Exclusion: $|A\cup B|=|A|+|B|-|A\cap B|$ $1450000=650000+125000-|A\cap B|$ $1450000=1900000-|A\cap B|$ $|A\cap B|=450000$