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

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

Q: How do I access the Complete Solution?

You can access the Complete Solution through our app, which can be downloaded using this link: App Link Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Q: Is this Complete Solution for IGNOU Assignments?

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Q: Is the Complete Solution aligned with the IGNOU requirements?

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Q: Is the Complete Solution guaranteed error-free?

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

Q: Can I access the Complete Solution anytime?

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

Q: What if I need help with a specific assignment question?

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

Q: Can I access the educational materials on multiple devices?

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Q: How do I make a payment for accessing the solutions?

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

Q: What are the instructions for formatting my assignments?

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.

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

₹365.00

Get Instant Access

Access via our Android App Only

Please read the following points before ordering :

Assignment Question Paper
Sample Solution
Frequently Asked Questions (FAQs)
Terms and Conditions

Share with your Friends

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.

Get The AppClick Here

Frequently Asked Questions (FAQs)

How do I access the Complete Solution?

You can access the Complete Solution through our app, which can be downloaded using this link:

App Link

Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Is this Complete Solution for IGNOU Assignments?

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Is the Complete Solution aligned with the IGNOU requirements?

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Is the Complete Solution guaranteed error-free?

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

Can I access the Complete Solution anytime?

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

What if I need help with a specific assignment question?

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

Can I access the educational materials on multiple devices?

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

How do I make a payment for accessing the solutions?

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

What are the instructions for formatting my assignments?

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.

Terms and Conditions

The educational materials provided in the app are the sole property of the app owner and are protected by copyright laws.
Reproduction, distribution, or sale of the educational materials without prior written consent from the app owner is strictly prohibited and may result in legal consequences.
Any attempt to modify, alter, or use the educational materials for commercial purposes is strictly prohibited.
The app owner reserves the right to revoke access to the educational materials at any time without notice for any violation of these terms and conditions.
The app owner is not responsible for any damages or losses resulting from the use of the educational materials.
The app owner reserves the right to modify these terms and conditions at any time without notice.
By accessing and using the app, you agree to abide by these terms and conditions.
Access to the educational materials is limited to one device only. Logging in to the app on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Our educational materials are solely available on our website and application only. Users and students can report the dealing or selling of the copied version of our educational materials by any third party at our email address (abstract4math@gmail.com) or mobile no. (+91-9958288900).

In return, such users/students can expect free our educational materials/assignments and other benefits as a bonafide gesture which will be completely dependent upon our discretion.

Categories IGNOU Assignment Solution, M.Sc. MACS, M.Sc. MACS 2024 Tags "IGNOU, 2024, Academic Success, Assignment Solutions, IGNOU Academic Guides 2024, IGNOU Mathematics Assignments, IGNOU MMT-004 Solved Assignment 2024, M.Sc. MACS, M.Sc. MACS Assignment Solutions, Mathematics, MMT-004, MMT-004 Assignment Answers, Solved Assignment, study materials

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

IGNOU MMT-004 Assignment Question Paper 2024

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

MMT-004 Sample Solution 2024

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

Frequently Asked Questions (FAQs)

Terms and Conditions

All Policies

Account

Company

Affiliate Program

Shipping Policy

Privacy Policy

Return Policy

Terms & Conditions

Cancellation Policy

Registration & Security

Refund Policy

Copyright & DMCA Notice

Privacy Policy

Content Copyright

Dispute Resolution

Terms of Services

Contact Us

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

IGNOU MMT-004 Assignment Question Paper 2024

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

MMT-004 Sample Solution 2024

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

Frequently Asked Questions (FAQs)

Terms and Conditions

Related products

IGNOU MEG-16 Solved Assignment 2022-2023 | MEG | Indian Folk Literature

IGNOU MEG-15 Solved Assignment 2022-2023 | MEG | Comparative Literature: Theory and Practice

IGNOU MEG-09 Solved Assignment 2022-2023 | MEG | Australian Literature

IGNOU MEG-05 Solved Assignment 2022-2023 | MEG | LITERARY CRITICISM & THEORY

All Policies

Account

Company

Contact Us