Chapter 9.5, Problem 68E

$p\left(n\right)$denote the number of different equivalence relations on a set withnelements (and byTheorem 2the number of partitions of a set withnelements). Show that $p\left(n\right)$satisfies the recurrence relation $p\left(n\right)={\displaystyle {\sum }_{j=0}^{n-1}C\left(n-1,j\right)p\left(n-j-1\right)}$and the initial condition $p\left(0\right)=1$. (Note: The numbers $p\left(n\right)$are called Bell numbers after the American mathematician E. T. Bell.)