\[\begin{gathered} {\text{Given the set }}X = \{ 1,2,3\} {\text{ with the discrete metric defined by }}d(a,b) = 1{\text{ if }}a \ne b{\text{ and }}d(a,a) = 0{\text{, what is the distance between 1 and 3?}} \hfill \\ \hfill \\ \end{gathered} \]

• A) 3
• B) 1
• C) 2
• D) 0

The order of the cyclic permutation (1 2 3 4 5 6 7) is -----------------

• A) 8
• B) 5
• C) 6
• D) 7

\[{\text{Which of the following points is at a taxicab distance of 4 from the point }}\left( {2,3} \right) \in {R^2}{\text{?}}\]

• A) \[\left( {{\text{5,5}}} \right)\]
• B) \[\left( {{\text{2,7}}} \right)\]
• C) \[\left( {{\text{0,2}}} \right)\]
• D) \[\left( {{\text{0,0}}} \right)\]

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

{{\text{The points }}\left( {{x_1},{x_2}, \ldots ,{x_n}} \right),\left( {{y_1},{y_2}, \ldots ,{y_n}} \right) \in {R^n}{\text{ have a taxicab distance of 0 if:}}}

• A) {x_i} = y{_i}{\text{ }}for{\text{ }}all{\text{ }}i
• B) {{x_i} = y{_i}{\text{ }}for{\text{ }}some{\text{ }}i}
• C)
• D)

{{\text{What is the taxicab distance between the points }}\left( {{\text{1,2}}} \right){\text{ and }}\left( {4,6} \right){\text{ in }}{R^2}{\text{?}}}

• A) 6
• B) 7
• C) 5
• D) 8

For the distance function d(x,y) = \sqrt {\left| {x - y} \right|} in R ,d(1,\frac{1}{2}) =-----------.

• A) 2
• B) 1
• C) \sqrt 2
• D) \frac{1}{{\sqrt 2 }}

{\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) 6
• B) 7
• C) 5
• D) 8

\begin{gathered} {\text{Given the set }}X = \{ 1,2,3\} {\text{ with the discrete metric defined by }}d(a,b) = 1{\text{ if }}a \ne b{\text{ and }}d(a,a) = 0{\text{, what is the distance between 1 and 3?}} \hfill \\ \hfill \\ \end{gathered}

• A) 2
• B) 3
• C) 0
• D) 1

{{\text{What is the taxicab distance between the points }}\left( {{\text{2,3}}} \right){\text{ and }}\left( {{\text{5,7}}} \right){\text{ in }}{R^2}{\text{?}}}

• A) 5
• B) 8
• C) 6
• D) 7

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

• A) 8
• B) 7
• C) 6
• D) 5

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

• A) 7
• B) 8
• C) 6
• D) 5

The order of (a b c) is ----------

• A) 2
• B) 4
• C) 3
• D) 1

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

• A) 6
• B) 5
• C) 8
• D) 7

Let $\alpha $ be a permutation such that ${\alpha ^7}\,\, = \,\,I$ , then order of $\alpha $ is ------------

• A) 5
• B) 6
• C) 7
• D) 8

\[{{\text{For points }}\left( {{x_1},{x_2}, \ldots ,{x_n}} \right){\text{ and }}\left( {{y_1},{y_2}, \ldots ,{y_n}} \right){\text{ in }}{R^n}{\text{, the taxicab distance is given by:}}}\]

• A) \[\sum\limits_{i = 1}^n {{{({x_i} - {y_i})}^2}} \]
• B) \[{\sum\limits_{i = 1}^n {({x_i} - {y_i})} }\]
• C) \[{\sum\limits_{i = 1}^n {\mid {x_i} - {y_i}} \mid }\]
• D) \[{max_{i = 1}^n\mid {x_i} - {y_i}}\]

The product of two odd permutations is an even permutation.

• A) False
• B) True
• C)
• D)

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

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

The order of a transposition is ----------

• A) 3
• B) 2
• C) 4
• D) 1

{{\text{Under the taxicab metric on }}{R^2}{\text{, when is the distance between two points }}\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right){\text{ equal to zero?}}}

• A) {\text{When }}{x_1} = y{_1}{\text{ and }}{x_2} = {y_2}
• B) {\text{When }}{x_1} = {y_2}{\text{ and }}{{\text{x}}_1} = {y_2}
• C) {\text{When }}{x_1} = {x_2}{\text{ and }}{y_1} = {y_2}
• D) {\text{None of these}}