Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$
I wish to use the Euclidean dissimilarity and Average linkage to calculate a dissimilarity matrix for these clusters.
So I use the formula $\frac{1}{|A||B|}\sum_{x\in A}\sum_{y\in B}\sqrt{\sum_{k=1}^{m}(x_{ik} - y_{jk})^2}$
But if I wish to calculate the dissimilarity using this measure between say $A$ and itself, I'm inclined to say from a rudimentary knowledge of metric spaces (and a basic intuition that $A$ should have $0$ dissimilarity with itself) that $d(A,A)=0$ for any $A$. However, using the formula given above I get $d(A,A)=0.5$;
$$\frac{1}{4}\left[\sqrt{(1-2)^2 + 0} + \sqrt{(2-1)^2 + 0} \right] = 0.5$$
Can someone reason with me which of the two answers is correct? I feel the conclusion probably hinges on whether $A$ is interpreted as a single entity or as a collection of sets.