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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.

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    When editing the question the linebreaks implied that this is somehow copy-pasted from somewhere else. This feels more so as $S(x,y)$ is not defined in the question given here.2011-10-28
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    See [Hamming distance](http://en.wikipedia.org/wiki/Hamming_distance).2011-10-28

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