Metric Spaces Quiz � Understand the Axioms and Properties of Metrics (MP.2, MP.7)

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

True

False

Symmetry property.

Explanation

Symmetry property.

2) D(x,y)=(x−y)² satisfies all metric properties.

True

False

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

Explanation

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

3) The discrete metric makes every subset closed.

True

False

Every set is both open and closed in discrete metric.

Explanation

Every set is both open and closed in discrete metric.

4) Which of these is an ultrametric?

D(x,y)=|x−y|

D(x,y)=0 if x=y, else 1

D(x,y)=‖x−y‖₂

D(x,y)=‖x−y‖₁

Discrete metric satisfies ultrametric inequality.

Explanation

Discrete metric satisfies ultrametric inequality.

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

D(x,y)=|x−y|

D(x,y)=0 if floor(x)=floor(y), else 1

D(x,y)=‖x−y‖₂

D(x,y)=‖x−y‖∞

Different points may have zero distance.

Explanation

Different points may have zero distance.

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

D(x,y)=‖x−y‖₁

D(x,y)=‖x−y‖₂

D(x,y)=‖x−y‖∞

Discrete metric

Maximum norm metric.

Explanation

Maximum norm metric.

7) All norms induce metrics.

True

False

Norms define metrics by d(x,y)=‖x−y‖.

Explanation

Norms define metrics by d(x,y)=‖x−y‖.

8) Which is a valid metric?

D(x,y)=|x−y|/(1+|x−y|)

D(x,y)=x²+y²

D(x,y)=|x−y|−2

D(x,y)=0 always

Satisfies metric properties.

Explanation

Satisfies metric properties.

9) Which of these is not a metric space?

(ℝ,|x−y|)

(ℝ,discrete metric)

(ℝ,d(x,y)=x+y)

(ℝ²,Euclidean)

x+y does not satisfy axioms.

Explanation

x+y does not satisfy axioms.

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

Symmetry

Triangle inequality

D(x,y)=0 ⇔ x=y

All of the above

Metrics don’t require linearity or norms.

Explanation

Metrics don’t require linearity or norms.

11) Every metric induces a topology.

True

False

Open balls form a topology.

Explanation

Open balls form a topology.

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

Identity

Symmetry

Triangle inequality

Non-negativity

Fails identity: distinct points have 0 distance.

Explanation

Fails identity: distinct points have 0 distance.

13) Which property does every metric satisfy?

Non-negativity

Linearity

Differentiability

Smoothness

Metrics must be non-negative.

Explanation

Metrics must be non-negative.

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

Non-negativity

Symmetry

Identity

Triangle inequality

Can yield negative values.

Explanation

Can yield negative values.

15) Which of these defines a metric on ℝ?

D(x,y)=|x−y|

D(x,y)=(x−y)²

D(x,y)=x+y

D(x,y)=√|x−y|

All except x+y are valid metrics.

Explanation

All except x+y are valid metrics.

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Counterexamples and Applications of Metrics

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

2)

What first name or nickname would you like us to use?

2) D(x,y)=(x−y)² satisfies all metric properties.

3) The discrete metric makes every subset closed.

4) Which of these is an ultrametric?

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

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

7) All norms induce metrics.

8) Which is a valid metric?

9) Which of these is not a metric space?

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

11) Every metric induces a topology.

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

13) Which property does every metric satisfy?

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

15) Which of these defines a metric on ℝ?