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MSCMACS 2026 - English

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Title Name	IGNOU MSCMACS MMT 4 Real Analysis Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 4
Subject Name	Real Analysis
Year	2026
Session	-
Language	English
Assignment Code	MMT 4/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 04 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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MMT 4 2026 - English

Assignment (MMT – 004)

Course Code: MMT-004
Assignment Code: MMT-004/TMA/2026
Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers. $5 \times 2 = 10$

a) The outer measure m^ of the set $A = \{x \in \mathbb{R} : x^2 = 1\} \cup [-3, 2]$ is 0.

b) A finite subset of a metric space is totally bounded.

c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.

d) The surface given by the equation $x + y + z - \sin(xyz) = 0$ can also be described by an equation of the form $z = f(x, y)$ in a neighbourhood of the point (0, 0).

e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].

2. a) Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

b) Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by

$$f(x, y, z) = (2x + 3y + z, xy, yz, xz)$

Find f(2, 0, -1).

c) Does Cantor’s intersection theorem hold for the metric space $X = (0, 1]$ with the standard metric? Justify your answer.

3. a) Obtain the second Taylor’s series expansion for the function given by

$$f(x, y) = xy^2 + 5xe^y \text{ at } \left(1, \frac{\pi}{2}\right).$

b) Find the Lebesgue integral of the function f given by

$$f = \chi_{[0, 1]} + 2\chi_{(3, 5]} + \chi_{[6, 9]}.$

c) Find and classify the extreme values of $(x, y) = xy$

Subject to the constraint

$$\frac{x^2}{8} + \frac{y^2}{2} - 1 = 0.$

Based on the image provided, here is the transcription of the text:

4. a) Let $f : \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}^3 \setminus \{(0,0,0)\}$ be given by

$$f(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2}, \frac{y}{x^2 + y^2 + z^2}, \frac{z}{x^2 + y^2 + z^2} \right)$

Show that f is locally invertible at all points in $\mathbb{R}^3 \setminus \{(0,0,0)\}$ .

b) For the equation $x^2 + y^2 + z^3 = 0$ , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form $z = f(x, y)$ .

c) Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

5. a) Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set $D \cup E$ where

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

b) Which of the following subsets of $\mathbb{R}$ are compact w.r.t. the metric given against them. Justify your answer.

i) $A = (0, 1)$ in $\mathbb{R}$ of $-\mathbb{R}$ with standard metric.
ii) $A = [3, 4] - \mathbb{R}$ with discrete metric.
iii) $\{(x, y) \in \mathbb{R}^2 \quad y > 0\} - \mathbb{R}^2$ with standard metric.

6. a) If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

b) Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

c) Find the interior and closure of the set $\mathbb{Q}$ of rationals in $\mathbb{R}$ with standard metric.

7. a) Let F be the function from $\mathbb{R}^2$ to $\mathbb{R}^2$ defined by

$$F(x, y) = (x^2 + y^2, xy).$

Show that F is differentiable at (1, 2). Find the differential matrix of F.

b) Show that the function f defined by
$$f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}$
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in $\mathbb{R}^2$ ? Justify your answer.

c) Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

8. a) Find the directional derivative of the function $f : \mathbb{R}^4 \to \mathbb{R}^4$ defined by

$$f(x, y, z, w) = (x^2y, xyz, x^2 + y^2, zw^2)$

at the point (1, 2, -1, -2) in the direction $v = (1, 0, -2, 2)$ .

b) Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

c) Find $B \left[ 2, \frac{1}{2} \right]$ in $\mathbb{R}$ where d is the metric given by $d(x, y) = \min\{1, |x - y|\}$ .

Based on the image provided, here is the transcription of the final questions:

9. a) Give an example of a family f_i of subsets of a set X which has finite intersection property. Justify your choice of example.

b) Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

10. a) Let (X, d) be a metric space and A be a subset of X. Show that $bdy(A) = \emptyset$ if and only if A is both open and closed.

b) Give an example of an algebra which is not a $\sigma$ -algebra. Justify your choice of examples.

c) If E is a measurable set and f is a simple function such that $a \leq f(x) \leq b \text{ } \forall x \in E$ , show that

$$am(E) \leq \int_{E} f \, dm \leq m(E)b.$

MMT 4 2025 - English

Course Code: MMT-004

Assignment Code: MMT-004/TMA/2025

Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers.

a) The outer measure m* of the set $A={xinmathbb{R}:x^2=1}cup{-3,2} ext{is}0.$

b) A finite subset of a metric space is totally bounded.

c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.

d) The surface given by the equation x+y+z-sin(xyz) = 0 can also be described by an equation of the form z = f(x, y) in a neighbourhood of the point (0,0).

e) A real valued function f on [a,b] is continuous if it is integrable on [a,b].

2. a) Find the interior, closure, the set of limit points and the boundary of the set

$A={(x,y)inmathbb{R}^2:y=1}$

in R² with the standard metric.

b) Consider $f:mathbb{R}^3 omathbb{R}^3 ext{given by}$

f(x,y,z) = (2x+3y+z,xy,yz,xz)

Find ƒ (2,0,-1).

c) Does Cantor's intersection theorem hold for the metric space X = (0,1] with the standard metric? Justify your answer.

3. a) Obtain the second Taylor's series expansion for the function given by

$f=chi_{[0,1]}+2chi_{[3,5]}+chi_{[6,9]}$

b) Find the Lebesgue integral of the function f given by

$f=chi_{[0,1]}+2chi_{[3,5]}+chi_{[6,9]}$

c) Find and classify the extreme values of (x, y) = xy Subject to the constraint

$frac{x^2}{8}+frac{y^2}{2}-1=0.$

4. a) $f:mathbb{R}^3setminus{(0,0,0)} omathbb{R}^3setminus{(0,0,0)}$ be given by

$f(x,y,z)=left(frac{x}{x^2+y^2+3^2},frac{y}{x^2+y^2+z^2},frac{z}{x^2+y^2+z^2} ight)$

Show that f is locally invertible at all points in $mathbb{R}^{3}$ {(0,0,0)}.

b) For the equation , x² + y³ + z³ = at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form z = f (x, y) .

c) Find the Fourier series of f (t) = t² on [−π,π].

5. a) Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set D U E where

$D={(x,y)inmathbb{R}^2:||(x+1,y)||=1} ext{and}$

$E={(x,y)inmathbb{R}^3:||x-1,y||=1}.$

b) Which of the following subsets of R are compact w.r.t. the metric given against them. Justify your answer.

i) A = (1,0) in $mathbb{R}$ of − $mathbb{R}$ with standard metric.

ii) A = [4,3] − $mathbb{R}$ with discrete metric.

iii) {(x, y) ∈ $mathbb{R}^{2}$ y > 0} − $mathbb{R}^{2}$ with standard metric

6. a) If E is a subset of $mathbb{R}$ with standard metric, then show that $(overline{E})^C=(E^C)^0.$

b) Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

c) Find the interior and closure of the set $mathbb{Q}$ of rationals in $mathbb{R}$ with standard metric.

7. a) Let F be the function from $mathbb{R}^{2}$ to $mathbb{R}^{2}$ defined by

F(x, y) = (x² + y² , xy)

Show that F is differentiable at (2,1) . Find the differential matrix of F.

b) Show that the function f defined by

$f(x,y)=egin{cases}frac{xy}{x^2+y^2},& ext{if}(x,y) e(0,0)\0,& ext{if}(x,y)=(0,0)end{cases}$

is not differentiable at (0,0). Does the partial derivatives of f exists at (0,0)? or any at any other point in R²? Justify your answer.

c) Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

8. a) Find the directional derivative of the function $f:mathbb{R}^{4} ightarrowmathbb{R}^{4}$ defined by

$f(x,y,z,w)=(x^2 y,xyz,x^2+y^2,zw^2)$

at the point (1,2,-1,-2) in the direction v = (1,0,-2,2).

b) Suppose that $f:mathbb{R} omathbb{R}^2$ is given by f(t) = (t,t²) and $g:mathbb{R}^2 omathbb{R}^3$ is given by g(x, y) = (x², xy, y²-x²). Compute the derivative of gof.

c) Find $Bleft[2,frac{1}{2} ight]$ in $mathbb{R}$ where d is the metric given by $d(x,y)=min{1,|x-y|}.$

9. a) Give an example of a family f₁ of subsets of a set X which has finite intersection property. Justify your choice of example.

b) Verify the hypothesis and conclusions of the Fatou's lemma for the sequence {f_n} given by

$f_n(x)=2n ext{for}xinleft(frac{1}{2n},frac{1}{n} ight)$

$=0quad ext{for}xinleft(0,frac{1}{2n} ight)cupleft(frac{1}{n},1 ight)$

10. a) Let (X,d) be a metric space and A be a subset of X. Show that bdy(A) = Qif and only if A is both open and closed.

b) Give an example of an algebra which is not a σ − algebra. Justify your choice of examples.

c) If E is a measurable set and f is a simple function such that a $ale f(x)le bquadforall xin E,$ , show that

$am(E)leint_E f,dmle m(E)b.$

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Course: M.Sc. Mathematics with Applications in Computer Science (MSCMACS)
Subject: MMT 4 - Real Analysis
Session: January 2026 - December 2026
Language: English

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Product Information

Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Subject Code	MMT 4
Subject	Real Analysis
Programme	MASTER DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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