Example 3 (i) Find the number of integers between 1 and 500 that are not divisible by any of 5, 6, 8;
(ii) Find the number of integers between 1 and 500 that are divisible by 5, or 6, or 8.
Consider the sequence $A$ composed of integers from 1 to 500. Property $P_{1}$: divisible by 5, property $P_{2}$: divisible by 6, property $P_{3}$: divisible by 8. Thus, (i) is to find the number of $B^{(0)}$, and (ii) is to find the number of $A^{(0)}-B^{(0)}$.
The number of $A(1)$ is: 100; the number of $A(2)$ is: 83; the number of $A(3)$ is: 62. $A(1,2)$ consists of integers divisible by 30, with a count of 16; $A(1,3)$ consists of integers divisible by 40, with a count of 12; $A(2,3)$ consists of integers divisible by 24, with a count of 20. $A(1,2,3)$ consists of integers divisible by 120, with a count of 4. Therefore,
$$\begin{array}{c}
B^{(0)}=500-(100+83+62)+(16+20+12)-(4)=299 \\
A^{(0)}-B^{(0)}=500-299=201
\end{array}$$