Metric Spaces Quiz (Advanced) � Explore Valid Metrics and Counterexamples (MP.2, MP.7)

1) For a metric d, d(x,x)=0 always holds.

True

False

Identity property.

Explanation

Identity property.

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

True

False

Symmetry property.

Explanation

Symmetry property.

3) In the discrete metric, d(x,y)=?

0 if x=y, 1 if x≠y

|x−y|

X+y

2|x−y|

Discrete metric definition.

Explanation

Discrete metric definition.

4) Triangle inequality states d(x,z) ≤ d(x,y)+d(y,z).

True

False

Triangle inequality.

Explanation

Triangle inequality.

5) The distance between any two points in a metric space is always non-negative.

True

False

Non-negativity property.

Explanation

Non-negativity property.

6) Which property is violated by d(x,y)=x−y on ℝ?

Non-negativity

Symmetry

Triangle inequality

Identity

d(1,2)=−1

Explanation

d(1,2)=−1

7) Which property distinguishes metrics from pseudometrics?

Symmetry

Triangle inequality

Non-negativity

D(x,y)=0 ⇒ x=y

Pseudometric may assign 0 to distinct points.

Explanation

Pseudometric may assign 0 to distinct points.

8) A metric on a set X is a function d: XmX → ℝ such that:

D(x,y) ≥ 0

D(x,y)=0 iff x=y

D(x,y)=d(y,x)

Triangle inequality holds

These are the four metric axioms.

Explanation

These are the four metric axioms.

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9) Which of the following is NOT a property of a metric?

Symmetry

Non-negativity

Identity of indiscernibles

Linearity

Linearity is not required.

Explanation

Linearity is not required.

10) In ℝ with d(x,y)=|x−y|, what is d(3,7)?

4

−4

10

7

|3-7|=4.

Explanation

|3-7|=4.

11) Which of these is a metric on ℝ²?

D(x,y)=‖x−y‖₂

D(x,y)=‖x−y‖₁

D(x,y)=‖x−y‖∞

All of the above

Euclidean, Manhattan, and Max norms are metrics.

Explanation

Euclidean, Manhattan, and Max norms are metrics.

12) Which function is NOT a metric?

D(x,y)=|x−y|

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

D(x,y)=x+y

D(x,y)=max{|x−y|,1}

x+y does not satisfy metric axioms.

Explanation

x+y does not satisfy metric axioms.

13) Which of the following is a metric on ℝ²?

D((x₁,y₁),(x₂,y₂))=|x₁−x₂|+|y₁−y₂|

D((x₁,y₁),(x₂,y₂))=|x₁−x₂|·|y₁−y₂|

D((x₁,y₁),(x₂,y₂))=√((x₁−x₂)²+(y₁−y₂)²)

Both a and c

Manhattan and Euclidean are metrics.

Explanation

Manhattan and Euclidean are metrics.

14) Which metric is called the Euclidean metric?

D(x,y)=|x−y|

D(x,y)=‖x−y‖₁

D(x,y)=‖x−y‖∞

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

Euclidean metric in ℝ.

Explanation

Euclidean metric in ℝ.

15) In a metric space, distance can be infinite.

True

False

Distance must be finite.

Explanation

Distance must be finite.

Properties of Metrics in Mathematics

1) For a metric d, d(x,x)=0 always holds.

2)

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

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

3) In the discrete metric, d(x,y)=?

4) Triangle inequality states d(x,z) ≤ d(x,y)+d(y,z).

5) The distance between any two points in a metric space is always non-negative.

6) Which property is violated by d(x,y)=x−y on ℝ?

7) Which property distinguishes metrics from pseudometrics?

8) A metric on a set X is a function d: XmX → ℝ such that:

9) Which of the following is NOT a property of a metric?

10) In ℝ with d(x,y)=|x−y|, what is d(3,7)?

11) Which of these is a metric on ℝ²?

12) Which function is NOT a metric?

13) Which of the following is a metric on ℝ²?

14) Which metric is called the Euclidean metric?

15) In a metric space, distance can be infinite.