IGNOU MMT-004 Solved Assignment 2024 for M.Sc. MACS

IGNOU MMT-004 Solved Assignment 2024 | M.Sc. MACS

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Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-004 Assignment Question Paper 2024

mmt-003-solved-assignment-2024-qp-2afdf4f2-c576-4553-8e87-8dfd8225b98e

MMT-003 Solved Assignment 2024 QP

a) Let $X = C [0, 1]$ . Define $d : X \times X \to R$ by $d (f, g) = \int_{0}^{1} | f (t) - g (t) | d t, f, g \in X$ where the integral is the Riemann integral. Show that $d$ is a metric on $X$ . Find $d (f, g)$ where $f (x) = 4 x$ and $g (x) = x^{3}, x \in [0, 1]$ .

b) Let (

X, d)

be a metric space and

a \in X

be a fixed point of

X

. Show that the function

f_{a} : X \to R

given by

f_{a} (x) = d (x, a)

is continuous. Is it uniformly continuous? Justify you answer.
2. a) Let

A

and

B

be any two subsets of a metric space (X, d), then show that
i) int

A = \cup {E

: is open and

E \subseteq A}

ii)

int (A \cap B) = int A \cap int B

iii)

int (A \cup B) \supseteq

int

A \cap

int

B

iv)

\bar{A \cap B} \subseteq \bar{A} \cap \bar{B}

.
b) Find the interior, boundary and closure of the following sets

A

R

with the usual metric and discrete metric.
i)

A = Q

, the set of rationals in

R

ii)

A =] 1, 2] \cup] 2, 4 [

3. a) Let

(X, d_{1})

and

(Y, d_{2})

be metric spaces. Show that

f : X \to Y

is continuous if and only if

f (\bar{A}) \subseteq \bar{f (A)}

where

A

is any subset of

X

b) Let

(X_{1}, d_{1})

and

(X_{2}, d_{2})

be two discrete metric spaces. Verify that the product metric on

X_{1} \times X_{2}

is discrete.
c) Show that an infinite discrete metric space

X

is bounded but not totally bounded.
4. a) Find the first derivative

f^{'} (a)

of the function

f

defined by

f : R^{3} \to R^{2}

given by

f (x, y, z) = (x y z, x + y + z^{2})

where

a = (1 . - 1, 2)

.
b) Let

E

be an open subset of

R^{n}

and

f : E \to R^{m}

be a function such that each of its components function

f_{i}

are differentiable, then show that

f

is differentiable. Is the converse of this result true? Justify your answer.
c) Near what points may the surface

z^{2} + x z + y = 0

be represented uniquely as a graph of a differentiable function

z = k (x, y)

? Locate such a point.
5. a) Use the method of Lagrange’s multiplier method to find the shortest possible distance from the ellipse

x^{2} + 2 y^{2} = 2

to the line

x + y = 2

.
b) Find the directional derivative of the function

f : R^{4} \to R^{3}

defined by

f (x, y, z, w) = (x^{2} y, x y z, x^{2} + y^{2} + z^{2})

a = (1, 2, - 1, - 2)

in the direction

v = (0, 1, 2, - 2)

.
6. a) Let A be a compact non-empty subset of a metric space (X, d) and let F be a closed subset of

X

such that

A \cap F = ϕ

, then show that

d (A, F) > 0

where

d (A, F) = inf {d (a, b) : a \in A, b \in F}

.
b) Give an example of the following with justification
i) A vector-valued function

f : R^{3} \to R^{3}

which is not differentiable at

(0, 0, 0)

.
ii) A function which is Legesgue measurable on

R

.
c) Show that the components of a metric space is either identical or pairwise disjoint.
7. a) Let

Q

be the set of rationals with the metric defined on

Q

d : Q \times Q \to R

, defined by

d (x, y) = | x - y |, \forall x, y \in R

.
Show that

{{(1 + \frac{1}{n})}^{n}}

is Cauchy sequence in

Q

, but does not converge in

Q

and

{\frac{1}{3^{n}}}

is a Cauchy sequence

Q

which converges in

Q

to the limit 0 .

b) Which of the following sets are totally bounded? Give reasons for your answer. Are they compact?
i)

2 N

(N, d)

where

d

is the discrete metric.
ii)

[0, 2] \cup [5, 10]

(R, d)

where

d

is the Euclidean metric.
c) Which of the following sets are connected sets in

R^{2}

with the metric given against it? Justify your answer.
i)

A = {(x, y) : 0 \leq x \leq 1, 0 \leq y \leq 2}

under the standard metric.
ii)

A = {(x, y) : x^{2} + y^{2} = 1}

under the discrete metric.
8. a) Consider

Z

and let

F_{1}

denote the class of subsets of

Z

, given by

F_{1} = {A \subset Z

: either

A

is finite or

A^{c}

is finite }. Check whether

F_{1}

is a

σ

algebra or not.
b) Let A be any set in

R

, show that

m^{*} (A) = m^{*} (A + x)

where

m^{*}

denotes the outer measure.
c) Find the measure of the following sets.

E = ⋂_{n = 1}^{\infty} (a - \frac{1}{n}, b)

ii)

E = Q \cup {1, 2, 3, 4}

iii)

E =] 5, 7 [\cup [7, 7.5]

.
9. a) Show that if

f

is measurable, then the function

f^{a} (x)

given by

f^{a} (x) = {\begin{array}{cc} a & if f (x) > a \\ f (x) & if f (x) \leq a \end{array}

is also measurable.
b) Verify Bounded Convergence Theorem for the sequence of functions

{f_{n}}

where

f_{n} (x) = \frac{1}{(1 + x / n)^{n}}, 0 \leq x \leq 1, n \in N

c) Find the fourier series of the function

f

defined by

f (x) = {\begin{matrix} - x^{2}, - π < x \leq 0 \\ x^{2}, 0 < x < π \end{matrix}

State whether the following statements are True or False. Justify your answers.
a) The sequence ${(\frac{1}{n}, \frac{1}{n}) : n \in N}$ is convergent in $R^{2}$ under the discrete metric on $R^{2}$ .
b) A subset in a metric space is compact if it is closed.
c) Continuous image of a path connected space is path connected.
d) The second derivative of a linear map from $R^{n}$ to $R^{m}$ never vanishes.
e) If $\int_{A} f d m = \int_{A} g d m$ for all $A \in M$ , then $f = g$ .

MMT-004 Sample Solution 2024

mmt-004-solved-assignment-2024-ss-020cab3d-1c01-486f-9bdf-7506d86b97ee

MMT-004 Solved Assignment 2024 SS

a) Let $X = C [0, 1]$ . Define $d : X \times X \to R$ by $d (f, g) = \int_{0}^{1} | f (t) - g (t) | d t, f, g \in X$ where the integral is the Riemann integral. Show that $d$ is a metric on $X$ . Find $d (f, g)$ where $f (x) = 4 x$ and $g (x) = x^{3}, x \in [0, 1]$ .

Answer:

To show that

d

is a metric on

X

, we need to verify the following properties for all

f, g, h \in X

Non-negativity: $d (f, g) \geq 0$
Identity of indiscernibles: $d (f, g) = 0$ if and only if $f = g$
Symmetry: $d (f, g) = d (g, f)$
Triangle inequality: $d (f, h) \leq d (f, g) + d (g, h)$

Let’s verify each property:

Non-negativity:
For any $f, g \in X$ , the absolute value function $| \cdot |$ ensures that $| f (t) - g (t) | \geq 0$ for all $t \in [0, 1]$ . Therefore, the integral $\int_{0}^{1} | f (t) - g (t) | d t$ is also non-negative. Hence, $d (f, g) \geq 0$ .
Identity of indiscernibles:
If $f = g$ , then $f (t) - g (t) = 0$ for all $t \in [0, 1]$ , so $d (f, g) = \int_{0}^{1} | f (t) - g (t) | d t = \int_{0}^{1} 0 d t = 0$ .
Conversely, if $d (f, g) = 0$ , then $\int_{0}^{1} | f (t) - g (t) | d t = 0$ . Since the integrand is non-negative, it must be zero almost everywhere, implying that $f (t) = g (t)$ for almost all $t \in [0, 1]$ . Since $f$ and $g$ are continuous, they must be equal everywhere on $[0, 1]$ , so $f = g$ .
Symmetry:
By the properties of the absolute value function, $| f (t) - g (t) | = | g (t) - f (t) |$ for all $t \in [0, 1]$ . Therefore, $d (f, g) = \int_{0}^{1} | f (t) - g (t) | d t = \int_{0}^{1} | g (t) - f (t) | d t = d (g, f)$ .
Triangle inequality:
For any $f, g, h \in X$ and for all $t \in [0, 1]$ , we have $| f (t) - h (t) | = | f (t) - g (t) + g (t) - h (t) | \leq | f (t) - g (t) | + | g (t) - h (t) |$ by the triangle inequality for real numbers. Integrating both sides over $[0, 1]$ gives:

$\int_{0}^{1} | f (t) - h (t) | d t \leq \int_{0}^{1} (| f (t) - g (t) | + | g (t) - h (t) |) d t = \int_{0}^{1} | f (t) - g (t) | d t + \int_{0}^{1} | g (t) - h (t) | d t = d (f, g) + d (g, h)$

Thus, $d (f, h) \leq d (f, g) + d (g, h)$ .

Since all four properties are satisfied,

d

is a metric on

X

Next, let’s find

d (f, g)

where

f (x) = 4 x

and

g (x) = x^{3}

for

x \in [0, 1]

d (f, g) = \int_{0}^{1} | 4 x - x^{3} | d x

To evaluate this integral, we can split it into regions where the integrand is positive or negative:

For $0 \leq x \leq 2$ , $4 x \geq x^{3}$ , so $| 4 x - x^{3} | = 4 x - x^{3}$ .
For $2 < x \leq 1$ , $4 x < x^{3}$ , so $| 4 x - x^{3} | = x^{3} - 4 x$ .

Thus, we can rewrite the integral as:

d (f, g) = \int_{0}^{2} (4 x - x^{3}) d x + \int_{2}^{1} (x^{3} - 4 x) d x

Now, we can calculate these integrals:

\begin{aligned} d (f, g) & = {[2 x^{2} - \frac{1}{4} x^{4}]}_{0}^{2} + {[\frac{1}{4} x^{4} - 2 x^{2}]}_{2}^{1} \\ = (2 (2)^{2} - \frac{1}{4} (2)^{4}) - (0 - 0) + (\frac{1}{4} (1)^{4} - 2 (1)^{2}) - (\frac{1}{4} (2)^{4} - 2 (2)^{2}) \\ = (8 - 4) + (\frac{1}{4} - 2) - (4 - 8) \\ = 4 - \frac{7}{4} - (- 4) \\ = \frac{16}{4} - \frac{7}{4} + \frac{16}{4} \\ = \frac{25}{4} \\ = 6.25 \end{aligned}

Therefore,

d (f, g) = 6.25

b) Let (

X, d)

be a metric space and

a \in X

be a fixed point of

X

. Show that the function

f_{a} : X \to R

given by

f_{a} (x) = d (x, a)

is continuous. Is it uniformly continuous? Justify you answer.

Answer:

To show that the function

f_{a} : X \to R

given by

f_{a} (x) = d (x, a)

is continuous, we need to show that for every

ϵ > 0

, there exists a

δ > 0

such that for all

x, y \in X

, if

d (x, y) < δ

, then

| f_{a} (x) - f_{a} (y) | < ϵ

Let

ϵ > 0

be given. Set

δ = ϵ

. Then, for any

x, y \in X

such that

d (x, y) < δ

, we have:

\begin{aligned} | f_{a} (x) - f_{a} (y) | & = | d (x, a) - d (y, a) | \\ \leq d (x, y) (by the triangle inequality) \\ < δ \\ = ϵ \end{aligned}

Hence,

f_{a}

is continuous.

Now, let’s consider whether

f_{a}

is uniformly continuous. A function is uniformly continuous if for every

ϵ > 0

, there exists a

δ > 0

such that for all

x, y \in X

, if

d (x, y) < δ

, then

| f_{a} (x) - f_{a} (y) | < ϵ

, where

δ

does not depend on the choice of

x

y

In metric spaces, continuity and uniform continuity are not always equivalent. However, if the metric space

X

is compact, then every continuous function from

X

R

is uniformly continuous. This is a consequence of the Heine-Cantor theorem.

If the metric space

X

is not compact, we cannot guarantee that

f_{a}

is uniformly continuous. For example, consider the metric space

(R, d)

where

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

is the standard metric on

R

. The function

f_{a} (x) = | x - a |

is continuous but not uniformly continuous on

R

In conclusion, the function

f_{a}

is continuous, but it is not necessarily uniformly continuous unless the metric space

X

has additional properties, such as compactness.

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IGNOU MMT-004 Solved Assignment 2024 | M.Sc. MACS

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IGNOU MMT-004 Solved Assignment 2024 | M.Sc. MACS

IGNOU MMT-004 Assignment Question Paper 2024

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