If d is a usual metric on$${\mathbb{R}^2}$$,then d((1,1),(1,2)) =-----------.

• A) 0
• B) 1
• C) 3
• D) 2

If d is a usual(natural) metric on R, then d(1,1)= ----------.

• A) 2
• B) 1
• C) 0.5
• D) 0

If$${d_0}$$ is a discrete metric on R, then$${d_0}( - 2, - 2) + {d_0}( - 2,2) = - - - - - .$$

• A) 2
• B) 0
• C) 1
• D) -1

$$In{\text{ }}a{\text{ }}metric{\text{ }}space{\text{ }}(X,d),{\text{ }}the{\text{ }}triangle{\text{ }}inequality{\text{ }}states{\text{ }}that{\text{ }}for{\text{ }}any{\text{ }}points{\text{ }}a,b,c \in X:$$

• A) $$d(a,c) \leqslant d(a,b) + d(b,c)$$
• B) $$d(a,b) \geqslant d(b,c)$$
• C) $$d(a,b) \leqslant d(b,c)$$
• D) $$d(a,c) = d(a,b) + d(b,c)$$

If d is a usual metric on R, then d(-1,1)= ----------.

• A) -1
• B) 2
• C) 1
• D) -2

$${\text{Determine if }}d\left( {x,y} \right) = \frac{{\mid x - y\mid }}{{1 + \mid x - y\mid }}{\text{ is a metric on R?}}$$

• A) $${\text{No}}$$
• B) $${\text{Yes}}$$
• C)
• D)

$${\text{If }}d\left( {a,b} \right) = \mid a - b\mid {\text{defines a metric on R, which of the following points are at distance 3 from the point 2?}}$$

• A) $${\text{ - 1 and 4}}$$
• B) $${\text{1 and - 4}}$$
• C) $${\text{2 and 5}}$$
• D) $${\text{ - 4 and 4}}$$

$${\text{What is the distance between the points }}\left( {{\text{1,2}}} \right){\text{ and }}\left( {4,6} \right){\text{ under usual or Eucledian metric on }}{R^2}{\text{ ?}}$$

• A) 6
• B) 7
• C) 8
• D) 5

$${\text{What is the distance between the points }}\left( {{\text{1,3}}} \right){\text{ and }}\left( {5,6} \right){\text{ under usual or Eucledian metric on }}{R^2}{\text{ ?}}$$

• A) 7
• B) 8
• C) 6
• D) 5

$${\text{In }}{{\text{R}}^5}{\text{, the taxicab distance between points }}\left( {1,2,0,3,4} \right){\text{ and}}\left( {0,4,1,5,6} \right){\text{ is:}}$$

• A) 8
• B) 7
• C) 6
• D) 5

If$${d_0}$$ is a discrete metric on R then $${d_0}( - 2,2) = - - - - - .$$

• A) 1
• B) 2
• C) 0
• D) -1

Euclidean metric on R is defined by …………

• A) $$d(x,y) = \left| x \right| - \left| y \right|$$
• B) $$d(x,y) = \sqrt {\left| {x - y} \right|}$$
• C) $$d(x,y) = \left| {x - y} \right|$$
• D) $$d(x,y) = \left| x \right| + \left| y \right|$$

Triangular inequality states that length of one side of a triangle is…………… than the sum of the length of other two sides.

• A) less
• B) greater
• C)
• D)

If d is a metric on R defined by$$d(x,y) = \left| x \right| + \left| y \right|$$ and d(x,y)=1, then ---------.

• A) x=y=1
• B) No conclusion
• C) x=0 and y=1
• D) x=1 and y=0

$${\text{For points }}\left( {{x_1},{y_1}} \right){\text{and }}\left( {{x_2},{y_2}} \right){\text{, the taxicab distance on }}{R^2}{\text{ is given by:}}$$

• A) $$\mid {x_1} - {x_2}\mid - \mid {y_1} - {y_2}\mid$$
• B) $$\mid {x_1} + {x_2}\mid - \mid {y_1} + {y_2}\mid$$
• C) $$\mid {x_1} + {x_2}\mid + \mid {y_1} + {y_2}\mid$$
• D) $$\mid {x_1} - {x_2}\mid + \mid {y_1} - {y_2}\mid$$

$$Which{\text{ }}of{\text{ }}the{\text{ }}following{\text{ }}is{\text{ }}an{\text{ }}example{\text{ }}of{\text{ }}a{\text{ }}metric{\text{ }}space?$$

• A) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}integers{\text{ }}with{\text{ }}the{\text{ }}usual{\text{ }}addition.$$
• B) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}real{\text{ }}numbers{\text{ }}with{\text{ }}the{\text{ }}Euclidean{\text{ }}distance{\text{ }}d\left( {a,b} \right) = \mid a - b\mid .$$
• C) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}complex{\text{ }}numbers{\text{ }}with{\text{ }}the{\text{ }}operation{\text{ }}of{\text{ }}multiplication.$$
• D) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}positive{\text{ }}integers{\text{ }}with{\text{ }}the{\text{ }}function{\text{ }}d\left( {a,b} \right) = a + b.$$

For the distance function $$d(x,z) = \left| {x - z} \right|$$in (R,d), $$d(x,z) \geqslant 0$$

• A) False
• B) True
• C)
• D)

Set of real numbers with distance function d(x,y)=|x-y| satisfy the reflexive property.

• A) True
• B) False
• C)
• D)

$${\text{Which of the following functions is a metric on R?}}$$

• A) $$d\left( {x,y} \right) = \mid x + y{\mid ^2}$$
• B) $$d\left( {x,y} \right) = \mid x + y\mid$$
• C) $$d\left( {x,y} \right) = \mid x\left| + \right|y\mid$$
• D) $$d\left( {x,y} \right) = \sqrt {\mid x - y\mid }$$

Set of real numbers with distance function d(x,y)=|x-y| satisfy the symmetric property.

• A) False
• B) True
• C)
• D)