Example 2.7 Among the integers from 1 to 1000, how many integers are divisible by 4 but not by 3 or 10?
Let $S=\{1,2, \cdots, 1000\}$. Suppose $s \in S$, if $s$ is divisible by $4,3,10$, then $s$ is said to have properties $a_{1}, a_{2}, a_{3}$, respectively. By Corollary 2.2, the number of integers sought is
$$
\begin{aligned}
N\left(a_{1} a_{2}^{\prime} a_{3}^{\prime}\right) & =N\left[a_{1}\left(1-a_{2}\right)\left(1-a_{3}\right)\right] \\
& =N\left(a_{1}-a_{1} a_{2}-a_{1} a_{3}+a_{1} a_{2} a_{3}\right) \\
& =N\left(a_{1}\right)-N\left(a_{1} a_{2}\right)-N\left(a_{1} a_{3}\right)+N\left(a_{1} a_{2} a_{3}\right) \\
& =\left[\frac{1000}{4}\right]-\left[\frac{1000}{4 \times 3}\right]-\left[\frac{1000}{4 \times 5}\right]+\left[\frac{1000}{4 \times 3 \times 5}\right] \\
& =250-83-50+16=133 .
\end{aligned}
$$