Why is this NP complete?
The algorithm you described isn't polynomial time because it has to consider every possible subset of customers (they are $2^m$ of these).

Solution

This problem is NP-hard because the vertex cover problem can be reduced to it.

https://en.wikipedia.org/wiki/Vertex_cover#Computational_problem

HINT: To do the reduction, consider how you would model $A$ as a graph and vice-versa. Once you've done that, consider how a "diverse set" is related to a vertex cover in the graphs.