Metric
Task number: 4030
Verify that the Hamming distance satisfies the axioms of the metric.
Hint
We use the definition \(\operatorname{dist}(x,y)=|\{i: x_i\ne y_i\}|\)
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\(\forall x,y\in \Sigma^n: \operatorname{dist}(x,y)\ge 0\)
… the cardinality of a finte set is a finite number -
\(\forall x,y\in \Sigma^n: \operatorname{dist}(x,y)= 0 \Longleftrightarrow x=y \)
… \(\{i: x_i\ne y_i\}=\emptyset\) if and only if \(x,y\) are identical -
\(\forall x,y\in \Sigma^n: \operatorname{dist}(x,y)=\operatorname{dist}(y,x)\)
… \(x_i\ne y_i\) if and only if \(y_i\ne x_i\) -
\(\forall x,y,z\in \Sigma^n: \operatorname{dist}(x,z)\le \operatorname{dist}(x,y)+\operatorname{dist}(y,z)\)
… \(x_i\ne z_i \Longrightarrow x_i\ne y_i \lor y_i\ne z_i\) and hence \( \{i: x_i\ne z_i\}\subseteq \{i: x_i\ne y_i\} \cup \{i: y_i\ne z_i\}\)
-
\(\forall x,y\in \Sigma^n: \operatorname{dist}(x,y)\ge 0\)
