Question

Let $S$ be the set of integers which are divisible by 5 , and let $T$ be the set of integer which are divisible by 7. Find the number of positive integers less than 1000 and not in ( $S \cup T$ ).

Answer 1

$n ( S )=5+10+\ldots \ldots+1000=200-1=199 $
$n ( T )=7+14+\ldots \ldots+994=142 $
$n ( S \cap T )=35+70+\ldots \ldots+980=28 $
$n ( S \cup T )=341-28=313= n ( S )+ n ( T )- n ( S \cap T )$

$\text { Total }=999 $
$n (\overline{ S } \cap \overline{ T })=999-313=686$