Chapter 9, Problem 23SE

pose that $P_1=\left\{A_1,A_2,\mathrm{....}{A}_m\right\}$and $P_2=\left\{B_1,B_2,\mathrm{....}{B}_m\right\}$are both partitions of the setS. Show that the collection of nonempty subsets of the form $\overline{A_i\cap B_j}$is a partition of S that is a refinement of bothP₁andP₂(see the preamble to Exercise 49 of Section 9.5).