Counterexamples and Applications of Metrics

  • 12th Grade
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| Attempts: 13 | Questions: 15 | Updated: Dec 10, 2025
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1) D(x,y)=(x−y)² satisfies all metric properties.

Explanation

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

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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.

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2) The discrete metric makes every subset closed.

Explanation

Every set is both open and closed in discrete metric.

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3) Which of these is an ultrametric?

Explanation

Discrete metric satisfies ultrametric inequality.

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4) Symmetry means d(x,y)=d(y,x).

Explanation

Symmetry property.

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5) Which example shows a pseudometric but not a metric?

Explanation

Different points may have zero distance.

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6) Which metric is known as the “maximum metric”?

Explanation

Maximum norm metric.

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7) All norms induce metrics.

Explanation

Norms define metrics by d(x,y)=‖x−y‖.

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8) Which is a valid metric?

Explanation

Satisfies metric properties.

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9) Which of these is not a metric space?

Explanation

x+y does not satisfy axioms.

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

Explanation

Metrics don’t require linearity or norms.

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

Explanation

Fails identity: distinct points have 0 distance.

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12) Which property does every metric satisfy?

Explanation

Metrics must be non-negative.

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13) Every metric induces a topology.

Explanation

Open balls form a topology.

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14) Which of these defines a metric on ℝ?

Explanation

All except x+y are valid metrics.

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

Explanation

Can yield negative values.

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D(x,y)=(x−y)² satisfies all metric properties.
The discrete metric makes every subset closed.
Which of these is an ultrametric?
Symmetry means d(x,y)=d(y,x).
Which example shows a pseudometric but not a metric?
Which metric is known as the “maximum metric”?
All norms induce metrics.
Which is a valid metric?
Which of these is not a metric space?
Which property makes a space metric but not necessarily normed?
Which property is violated by d(x,y)=0 for all x,y?
Which property does every metric satisfy?
Every metric induces a topology.
Which of these defines a metric on ℝ?
Which property fails for d(x,y)=|x−y|−1?
Alert!