Exercises

Exercises Exercises

1.

Let

X

be a set. Show that the function

d

(the discrete metric) defined by

d (x, y) = {\begin{cases} 0 & if x = y \\ 1 & if x \neq y \end{cases}

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is a metric.

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2.

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Let

X = {1, 3, 5} \subset Z

and define

d : X \times X \to R

d (x, y) = x y - 1 (\mod n) .

That is,

d (x, y)

is the remainder when

x y - 1

is divided by

n .

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For each value of $n,$ determine if $d$ defines a metric on $X .$ Prove your answers.
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The unit circle in $R^{2}$ with metric $d$ is the set of all points in $R^{2}$ whose distance from the origin is $1 .$ If we let the distance be less than $1,$ then we have what we call an open ball. We can make this same definition in any metric space.
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Definition 3.11.

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Let $(Y, d_{Y})$ be a metric space. For any positive real number $r,$ the open ball centered at $b$ of radius $r$ in $(Y, d_{Y})$ is the the set

$B (b, r) = {y \in Y ∣ d_{Y} (y, b) < r} .$

If $(X, d)$ is a metric space for a given value of $n,$ determine all of the open balls in $X$ centered at $1 .$ If $(X, d)$ is not a metric space, explain why.

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(a)

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n = 4

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(b)

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n = 8

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3.

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Let

Q

be the set of all rational numbers in reduced form. A rational number

\frac{r}{s}

is in reduced form if

s > 0

and

r

and

s

have no common factors larger than

1 .

Define

d : Q \times Q \to R

d (\frac{a}{b}, \frac{r}{s}) = max {| a - r |, | b - s |} .

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(a)

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Prove that

d

is a metric.

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(b)

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A metric allows us to determine which elements in our metric space are ’’close’’ together. Describe the set of elements in

Q

that are a distance no more than

1

from

\frac{2}{3}

using this metric

d .

In other words, describe the open ball centered at

\frac{2}{3}

with radius

1

(see Definition 3.11).

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(c)

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a,

b,

and

c

are elements of a metric space

(X, d_{X}),

we say that

b

is between

a

and

c

d_{X} (a, c) = d_{X} (a, b) + d_{X} (b, c) .

Using the Euclidean metric on

Q,

there are infinitely many different rational numbers between

0

and

1

(the rational numbers between

0

and

1

that lie on the Euclidean line through

0

and

1 .

Describe all of the points in

(Q, d)

that are between

0

and

1 .

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4.

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Let

(Q, d)

be the metric space from Exercise 3. If

a,

b,

and

c

are elements of a metric space

(X, d_{X}),

we say that

b

is between

a

and

c

d_{X} (a, c) = d_{X} (a, b) + d_{X} (b, c) .

Using the Euclidean metric on

Q,

there are infinitely many different rational numbers between

0

and

1

(the rational numbers between

0

and

1

that lie on the Euclidean line through

0

and

1 .

In this exercise we explore numbers that are between others in the space

(Q, d) .

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(a)

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Find all of the elements in

(Q, d)

that are between

0

and

1 .

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(b)

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Which is closer to

0

(Q, d) :

1

\frac{1}{3} ?

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(c)

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Now find all of the elements in

(Q, d)

that are between

0

and

\frac{1}{3} .

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5.

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Prove that the taxicab metric

d_{T}

is a metric on

R^{n} .

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6.

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Let

A

and

B

be nonempty finite subsets of

R^{n},

and let

A + B = {a + b ∣ a \in A, b \in B} .

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(a)

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Prove that

max (A + B) \leq max A + max B .

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(b)

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Prove that the max metric

d_{M}

is a metric on

R^{n} .

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7.

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x = (x_{1}, x_{2}, \dots, x_{n}),

we let

| x | = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2}} .

For

x = (x_{1}, x_{2}, \dots, x_{n})

and

y = (y_{1}, y_{2}, \dots y_{n}),

define

d_{H} : R^{n} \times R^{n} \to R

d_{H} (x, y) = {\begin{cases} 0 & if x = y \\ | x | + | y | & otherwise . \end{cases}

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(a)

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Show that

d_{H}

is a metric (called the hub metric).

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(b)

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(i)

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Let

a = (\frac{1}{2}, 0) .

Explicitly describe which points are in the set

B (a, 1)

(R^{2}, d_{H}) .

(See Exercise 2 for the definition of an open ball.)

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(ii)

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Let

a = (3, 4) .

Explicitly describe which points are in the set

B (a, 1)

(R^{2}, d_{H}) .

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(iii)

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Now explicitly describe all open balls in

(R^{2}, d_{H}) .

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8.

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Let

Z

be the set of integers and let

p

be a prime. For each pair of distinct integers

m

and

n

there is an integer

t = t (m, n)

such that

| m - n | = k \times p^{t},

where

p

does not divide

k .

For example, if

p = 5,

m = 34,

and

n = 7,

then

m - n = 27 = 27 \times 5^{0} .

t (43, 7) = 27 .

However, if

m = 54

and

n = 4,

then

m - n = 50 = 2 \times 5^{2} .

t (54, 4) = 2 .

Define

d : Z \times Z \to R

d (m, n) = {\begin{cases} 0 & if m = n \\ \frac{1}{p^{t}} & if m \neq n . \end{cases}

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(a)

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Determine the values of

d (62, 170)

using

p = 3

and

d (14008, 2003)

using

p = 7 .

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(b)

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Prove that if

a,

b,

and

c

are in

Z,

then

t (a, c) \geq min {t (a, b), t (b, c)} .

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(c)

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Prove that

(Z, d)

is a metric space.

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(d)

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Let

p = 3 .

Describe the set of all elements

x

(Z, d)

such that

d (x, 0) = 1 .

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(e)

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Continue with

p = 3 .

Describe the set of all elements

x

(Z, d)

such that

d (x, 0) < \frac{1}{2} .

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9.

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Let

(X, d_{X})

and

(Y, d_{Y})

be metric spaces. We can make the Cartesian product

X \times Y

into a metric space by defining a metric

d^{'}

X \times Y

as follows. If

(x_{1}, y_{2})

and

(x_{2}, y_{2})

are in

X \times Y,

then

d^{'} ((x_{1}, y_{1}), (x_{2}, y_{2})) = max {d_{X} (x_{1}, x_{2}), d_{Y} (y_{1}, y_{2})} .

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You may assume without proof that

d^{'}

is a metric on

X \times Y .

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(a)

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Let

(X, d_{X}) = (R^{2}, d_{M})

and

(Y, d_{Y}) = (R^{2}, d_{T}) .

Let

u = ((1, 2), (1, - 1))

and

v = ((0, 5), (2, - 2)) .

What is

d^{'} (u, v) ?

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Recall that

d_{M} ((x_{1}, x_{2}), (y_{1}, y_{2})) = max {| x_{1} - y_{1} |, | x_{2} - y_{2} |}

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and

d_{T} ((x_{1}, x_{2}), (y_{1}, y_{2})) = | x_{1} - y_{1} | + | x_{2} - y_{2} | .

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(b)

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Let

(X, d_{X}) = (R, d_{E})

and

(Y, d_{Y}) = (R, d),

where

d

is the discrete metric. Let

A = {(x, y) \in R \times R ∣ - 1 \leq x \leq 1 and - 1 \leq y \leq 1}

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X \times Y .

Let

a = (0, 1)

X \times Y .

Describe, geometrically, what the open ball

B (a, 1)

looks like in the product space

X \times Y .

Draw a picture of this open ball.

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10.

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Let

X = R^{+},

the set of positive reals, and define

d : X \times X \to R

d (x, y) = | \ln (y / x) | .

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d

is a metric on

X ?

Prove your answer.

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11.

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Let

d : R \times R \to R

be defined by

d (x, y) = \frac{| x - y |}{| x - y | + 1} .

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Show that

d

is a metric on

R .

(Hint: For the triangle inequality, note that

d (x, y) = f (| x - y |)

where

f (t) = \frac{t}{t + 1},

and

f

is an increasing function.)

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12.

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Let

(X, d)

be a metric space and

k

be a constant. Define

k d : X \times X \to R

(k d) (x, y) = k d (x, y) .

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Under what, if any, conditions is

k d

a metric on

X .

Justify your answer.

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13.

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A real valued function

f

on an interval is concave if

\begin{matrix} (3.3) & f ((1 - α) x + α y) \geq (1 - α) f (x) + α f (y) \end{matrix}

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for all

α \in [0, 1]

and all

x

and

y

in the interval. Note that the expression

(1 - α) x + α y

is linear in

α

and is equal to

x

when

α = 0

and

y

when

α = 1 .

(1 - α) x + α y

is a parameterization of the line segment joining

x

y .

As Figure 3.12 indicates, equation (3.3) implies that the graph of a concave function

f

on any interval

[x, y]

lies above the secant line joining the points

(x, f (x))

and

(y, f (y)) .

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(a)

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Let

f (x) = - x^{2}

map

R

R

with the standard Euclidean metric. Show that

f

is concave on the interval

[- 1, 1] .

Hint.

Start with the fact that

α (1 - α) (x - y)^{2} \geq 0 .

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(b)

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Show that if

f

is a concave function on

[0, \infty)

and

f (0) \geq 0,

an interval and

a

and

b

are in the interval, then

f (a) + f (b) \geq f (a + b) .

Hint.

Consider (3.3) with

y = 0 .

Then use the fact that

\frac{a}{a + b}

is in

[0, 1] .

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(c)

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Suppose

(X, d)

is a metric space and

f : [0, \infty) \to [0, \infty)

is an increasing, concave function such that

f (x) = 0

if and only if

x = 0 .

Prove that

f \circ d

is a metric on

X .

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14.

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For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.

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(a)

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The function

d : R \times R \to R

defined by

d (x, y) = (x - y)^{2}

is a metric on

R .

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(b)

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Every nonempty set can be made into a metric space.

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(c)

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It is possible to define an infinite number of metrics on every set containing at least two elements.

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(d)

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Let

(X, d_{X})

and

(Y, d_{Y})

be metric spaces with

| X | \geq 2 .

Then the function

d : X \times Y \to R

defined by

d ((a, b), (c, d)) = d_{X} (a, c) d_{Y} (b, d)

is a metric on

X \times Y .

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(e)

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Let

(X, d)

be a metric space. If

X

is infinite, then the range of

d

is also an infinite set.

Exercises Exercises

1.

2.

Definition 3.11.

(a)

(b)

3.

(a)

(b)

(c)

4.

(a)

(b)

(c)

5.

6.

(a)

(b)

7.

(a)

(b)

(i)

(ii)

(iii)

8.

(a)

(b)

(c)

(d)

(e)

9.

(a)

(b)

10.

11.

12.

13.

(a)

(b)

(c)

14.

(a)

(b)

(c)

(d)

(e)