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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:xiyi}|

    • x,yΣn:dist(x,y)0
      … the cardinality of a finte set is a finite number
    • x,yΣn:dist(x,y)=0x=y
      {i:xiyi}= if and only if x,y are identical
    • x,yΣn:dist(x,y)=dist(y,x)
      xiyi if and only if yixi
    • x,y,zΣn:dist(x,z)dist(x,y)+dist(y,z)
      xizixiyiyizi and hence {i:xizi}{i:xiyi}{i:yizi}
Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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