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Which of the following is NOT an axiom that the distance function in a metric space must satisfy? Od(x,z) = d(x,y) + d(y,z) Od(x,y) = d(y,x) Od(x,y) = 0 iff x = y O d(x,z) ≤d(x,y) + d(y,z)

Which of the following is NOT an axiom that the distance function in a metric space must satisfy? Od(x,z) = d(x,y) + d(y,z) Od(x,y) = d(y,x) Od(x,y) = 0 iff x = y O d(x,z) ≤d(x,y) + d(y,z)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question:** Which of the following is NOT an axiom that the distance function in a metric space must satisfy? - ⭕ \(d(x,z) = d(x,y) + d(y,z)\) - ⭕ \(d(x,y) = d(y,x)\) - ⭕ \(d(x,y) = 0 \text{ iff } x = y\) - ⭕ \(d(x,z) \leq d(x,y) + d(y,z)\) **Explanation:** This question examines your understanding of the properties that define a metric in a metric space. A metric must satisfy the following axioms: 1. **Non-negativity:** \(d(x,y) \geq 0\) and \(d(x,y) = 0 \text{ iff } x = y\). 2. **Symmetry:** \(d(x,y) = d(y,x)\). 3. **Triangle Inequality:** \(d(x,z) \leq d(x,y) + d(y,z)\). The option \(d(x,z) = d(x,y) + d(y,z)\) is NOT an axiom required for a distance function in a metric space. This suggests equality rather than the inequality expected in the triangle inequality.
Transcribed Image Text:**Question:** Which of the following is NOT an axiom that the distance function in a metric space must satisfy? - ⭕ \(d(x,z) = d(x,y) + d(y,z)\) - ⭕ \(d(x,y) = d(y,x)\) - ⭕ \(d(x,y) = 0 \text{ iff } x = y\) - ⭕ \(d(x,z) \leq d(x,y) + d(y,z)\) **Explanation:** This question examines your understanding of the properties that define a metric in a metric space. A metric must satisfy the following axioms: 1. **Non-negativity:** \(d(x,y) \geq 0\) and \(d(x,y) = 0 \text{ iff } x = y\). 2. **Symmetry:** \(d(x,y) = d(y,x)\). 3. **Triangle Inequality:** \(d(x,z) \leq d(x,y) + d(y,z)\). The option \(d(x,z) = d(x,y) + d(y,z)\) is NOT an axiom required for a distance function in a metric space. This suggests equality rather than the inequality expected in the triangle inequality.
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