please prove if these two methods of computing cluster distance are equivalent:

i. Calculate the cluster centroids (mean point of all points in a cluster) and merge the two clusters with the closest centroids

ii. Calculate the average pairwise distance between all points across pairs of clusters, and merge those clusters with the smallest average pairwise distance.

If the two are equivalent, prove it mathematically. If the two are different, supply a (simple) case where the two measures will select different clusters to merge, give the average pairwise distances and cluster centroid distances, and show a plot of your constructed dataset.