Definition of Hamming Distance

Given a $q$-ary alphabet $A=\{a_1,a_2,\dots ,a_i\}$, the Hamming distance $d(w_1,w_2)$ between two words $w_1,w_2\in A^n$ is the total number of positions in which $w_1$ has a different symbol than $w_2$.

Rewriting with mathematical symbols, that means:

$$d(w_1,w_2):=|\{i\in \{1,2,\dots ,n\}:x_i\ne y_i\}|,\text{ where}\{\begin{array}{l}{w}_1=(x_1,x_2,\dots ,x_i)\in A^n\\ w_2=(y_1,y_2,\dots ,y_i)\in A^n\end{array}$$