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Let $X$ be the set of all ordered triples of zeros and ones. Show that $X$ consists of eight elements and a metric $d$ on $X$ is defined by $$d(x, y) = \text{number of places where}~~ x~~ \text{and}~~ y ~~\text{have different entries}$$

Mikasa
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user96395
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1 Answers1

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More generally, $\mathbb R^n$ is a metric space with the metric $d(x,y)=\sum_{k=1}^n |x_k-y_k|$. The triangle inequality easily follows from the triangle inequality for real numbers.

Restricting the above to the set $\{0,1\}^n\subset \mathbb R^n$ yields the space described in the problem.

The fact that the cardinality of $\{0,1\}^n$ is $2^n$ can be proved by induction: $\{0,1\}^n $ is the disjoint union of $\{0\}\times \{0,1\}^{n-1} $ and $\{1\}\times \{0,1\}^{n-1} $.