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GATE 2021 MA – Question 21

Real Analysis · Metric Spaces and Continuity · 1 mark · Multiple choice

On (0,1): P:d₁(x,y)=|log(x/y)| is a metric. Q:d₂(x,y)=|x|+|y| for x≠y and zero for x=y is a metric. Choose truth values.

  1. Both P and Q true
  2. P true, Q false
  3. P false, Q true
  4. Both false

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Show answer and explanation

Correct answer: (A) Both P and Q true

Explanation

d₁=|log x−log y| transports the ordinary metric by the injective map log. For d₂, positivity and symmetry hold; when an intermediate point differs from both endpoints, d₂(x,z)+d₂(z,y)=x+y+2z≥x+y=d₂(x,y). If it equals an endpoint the triangle inequality is equality. Both are metrics. **A**.

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