GATE 2026 MA – Question 20

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

Let d₁,d₂,d₃ be metrics on a set X with at least two elements. Which formula does NOT define a metric?

$\min(d_1(x,y),3)$
$\max(d_2(x,y),3)$
$10d_3(x,y)/(1+d_3(x,y))$
$(d_1(x,y)+d_2(x,y)+d_3(x,y))/3$

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Show answer and explanation

Correct answer: (B) $\max(d_2(x,y),3)$

Explanation

For B, d(x,x) = max(0,3) = 3, violating zero self-distance. Truncation, positive averaging and the increasing subadditive transform 10t/(1+t) preserve metric properties. **B**.

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