Discussion
For \(x, y \in \mathbf{R}\), define
\[ d_{1}(x, y) = \big|\, |x| - |y| \,\big| \]
and \(d_{2}(x, y) = |x^{2} - y^{2}|\). Then :
Answer: D
Neither \(d_{1}\) nor \(d_{2}\) is a metric on R.
A metric must vanish only on equal points. Both functions see only \(|x|\):
\[ \begin{aligned} d_{1}(1, -1) &= \big|\,|1| - |-1|\,\big| = 0 \\ d_{2}(1, -1) &= |1 - 1| = 0 \end{aligned} \]
yet \(1 \neq -1\). So both fail the identity of indiscernibles. (Each is symmetric and satisfies the triangle inequality, so each is a pseudometric; they become metrics only on \([0, \infty)\).)
That rules out every option claiming that one or both are metrics. The trap is that \(d_{1}\) looks like the familiar \(|x - y|\) and passes the triangle inequality check.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development