Does the following function define a distance metric?

Question

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is

$$\sum_1^N a_n \succ b_n$$

Note that this is very similar to the question asked here:

Is it wrong to use Binary Vector data in Cosine Similarity?

The difference is that the above question is for binary data, and the defined function performs a numeric comparison and sums the binary result.

Answer 1

The question about a vector of length one, i.e. the scalar case, made the answer clear: the comparison $(x \succ y) \ne (y \succ x)$ so the distance is not symmetric. It is not a distance metric.