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Metric
Task number: 4030
Verify that the Hamming distance satisfies the axioms of the metric.
Hint
We use the definition dist(x,y)=|{i:xi≠yi}|
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∀x,y∈Σn:dist(x,y)≥0
… the cardinality of a finte set is a finite number -
∀x,y∈Σn:dist(x,y)=0⟺x=y
… {i:xi≠yi}=∅ if and only if x,y are identical -
∀x,y∈Σn:dist(x,y)=dist(y,x)
… xi≠yi if and only if yi≠xi -
∀x,y,z∈Σn:dist(x,z)≤dist(x,y)+dist(y,z)
… xi≠zi⟹xi≠yi∨yi≠zi and hence {i:xi≠zi}⊆{i:xi≠yi}∪{i:yi≠zi}
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∀x,y∈Σn:dist(x,y)≥0