Metric property of simple matching dissimilarity ($1-S(x,y)=p^{-1}{\sum }_{j=1}^p{1}_{\{x_j\ne y_j\}}$)

The displayed distance is the normalized Hamming distance. Each coordinate mismatch between $x,z$ implies a mismatch between $x,y$ or $y,z$, so summing the indicator inequalities proves the triangle inequality. Nonnegativity, symmetry and separation follow immediately. It is a metric on profiles; on separately labelled individuals with identical profiles it is a pseudometric.