A DNA molecule can be represented as a string of symbols $A, C, G$ and $T$, such as $GGATAATTCTAG\ldots GACCGTACCC$.
For the purposes of this question, we will assume that all DNA molecules contain the same (large!) number $N$ of symbols. Thus, a DNA molecule is an $N$-tuple $x = (x_1,\ldots , x_N)$ where $x_i\in\{A,C,G,T\}$ for each $i$.
Define the distance between two DNA molecules $x, y$ as the number of elements $i$ in ${1, \ldots ,N}$ such that $x_i\neq y_i$.
Prove that this defines a metric on the set of DNA molecules.
Tutor defined a set $S(x,y)=\{1,3,6\}$. Why is this? The rest of the working after that is fine.