Counterexamples and Applications of Metrics

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7049 | Total Attempts: 9,519,298
| Questions: 15
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
1) Which of these defines a metric on ℝ?

Explanation

All except x+y are valid metrics.

Submit
Please wait...
About This Quiz
Counterexamples And Applications Of Metrics - Quiz

Not every “distance” is a metric! In this quiz, you’ll explore counterexamples and real applications of metric definitions. Try this quiz to sharpen reasoning in analysis.

2)
We’ll put your name on your report, certificate, and leaderboard.
2) D(x,y)=(x−y)² satisfies all metric properties.

Explanation

It is non-negative, symmetric, and satisfies triangle inequality.

Submit
3) Which property fails for d(x,y)=|x−y|−1?

Explanation

Can yield negative values.

Submit
4) Which metric is known as the “maximum metric”?

Explanation

Maximum norm metric.

Submit
5) All norms induce metrics.

Explanation

Norms define metrics by d(x,y)=‖x−y‖.

Submit
6) Which is a valid metric?

Explanation

Satisfies metric properties.

Submit
7) Which property is violated by d(x,y)=0 for all x,y?

Explanation

Fails identity: distinct points have 0 distance.

Submit
8) The discrete metric makes every subset closed.

Explanation

Every set is both open and closed in discrete metric.

Submit
9) Which of these is an ultrametric?

Explanation

Discrete metric satisfies ultrametric inequality.

Submit
10) Which property does every metric satisfy?

Explanation

Metrics must be non-negative.

Submit
11) Symmetry means d(x,y)=d(y,x).

Explanation

Symmetry property.

Submit
12) Which of these is not a metric space?

Explanation

x+y does not satisfy axioms.

Submit
13) Which property makes a space metric but not necessarily normed?

Explanation

Metrics don’t require linearity or norms.

Submit
14) Every metric induces a topology.

Explanation

Open balls form a topology.

Submit
15) Which example shows a pseudometric but not a metric?

Explanation

Different points may have zero distance.

Submit
View My Results

Quiz Review Timeline (Updated): Oct 13, 2025 +

Our quizzes are rigorously reviewed, monitored and continuously updated by our expert board to maintain accuracy, relevance, and timeliness.

  • Current Version
  • Oct 13, 2025
    Quiz Edited by
    ProProfs Editorial Team
  • Oct 07, 2025
    Quiz Created by
    Thames
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Which of these defines a metric on ℝ?
D(x,y)=(x−y)² satisfies all metric properties.
Which property fails for d(x,y)=|x−y|−1?
Which metric is known as the “maximum metric”?
All norms induce metrics.
Which is a valid metric?
Which property is violated by d(x,y)=0 for all x,y?
The discrete metric makes every subset closed.
Which of these is an ultrametric?
Which property does every metric satisfy?
Symmetry means d(x,y)=d(y,x).
Which of these is not a metric space?
Which property makes a space metric but not necessarily normed?
Every metric induces a topology.
Which example shows a pseudometric but not a metric?
Alert!

Back to Top Back to top
Advertisement