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IGNOU MMT 4 2024 Solution

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Last Date of Submission of IGNOU MMT-04 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MMT 4 Real Analysis Solved Assignment 2024
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	2024
Session	-
Language	English Medium
Assignment Code	MMT-04/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 04 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-04 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMT 4/2024

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Questions Included in this Help Book

Ques 1.

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

Ques 2.

Let $(X, d)$ be a metric space and $a \in X$ be a fixed point of $X$ . Show that the function $f_a : X \rightarrow R$ given by $f_a (x) = d(x,a)$ is continuous. Is it uniformly continuous? Justify you answer

Ques 3.

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) $\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$ .

Ques 4.

Find the interior, boundary and closure of the following sets A in $\mathbb{R}$ with the usual metric and discrete metric.

i) A = $\mathbb{Q}$ , the set of rationals in $\mathbb{R}$

ii) $A = ]1,2] \cup ]2 ,4[$

Ques 5.

Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. Show that $f : X \rightarrow Y$ is continuous if and only if $f (\overline{A}) \subseteq \overline{f(A)}$ where A is any subset of X

Ques 6.

Show that an infinite discrete metric space X is bounded but not totally bounded.

Ques 7.

Find the first derivative $f'(a)$ of the function f defined by $f : R^3 \rightarrow R^2$ given by $f(x,y,z) = (xyz, x + y+z^2)$ where $a = (1. -1,2)$ .

Ques 8.

Let E be an open subset of $R^n$ and $f : E \rightarrow R^m$ 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.

Ques 9.

Near what points may the surface $z^2 + xz + y = 0$ be represented uniquely as a graph of a differentiable function $z = k(x,y)$ ? Locate such a point.

Ques 10.

Use the method of Lagrange’s multiplier method to find the shortest possible distance from the ellipse $x^2 + 2y^2 = 2$ to the line $x + y = 2$ .

Ques 11.

Find the directional derivative of the function $f : R^4 \rightarrow R^3$ defined by $f(x, y, z, w) = (x^2y, xyz, x^2 + y^2 +zw^2)$ at $a = (1,2,-1,-2)$ in the direction $v = (0,1,2,-2)$ .

Ques 12.

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 $A \cap F = \phi$ , then show that $d(A, F)> 0$ where $d(A, F) = inf \left \{ d(a,b): a\in A, b \in F \right \}$ .

Ques 13.

Give an example of the following with justification
i) A vector-valued function $f : R^3 \rightarrow R^3$ which is not differentiable at $(0,0,0)$ .
ii) A function which is Legesgue measurable on R.

Ques 14.

Show that the components of a metric space is either identical or pairwise disjoint.

Ques 15.

Let Q be the set of rationals with the metric defined on Q by $d :Q\times Q \rightarrow R$ , defined by $d(x, y) = | x - y |, \forall x, y \in R$ . Show that $\left \{ \left ( 1 + \frac{1}{n}\right )^n \right \}$ is Cauchy sequence in Q, but does not converge in Q and $\left \{ \frac{1}{3^n} \right \}$ is a Cauchy sequence Q which converges in Q to the limit 0 .

Ques 16.

Which of the following sets are totally bounded? Give reasons for your answer. Are they compact?
i) 2N in (N,d) where d is the discrete metric.
ii) $[0,2] \cup [5,10]$ in (R,d) where d is the Euclidean metric.

Ques 17.

Which of the following sets are connected sets in 2 R 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) $A = \left \{ ( x,y) : x^2 + y^2 = 1\right \}$ under the discrete metric.

Ques 18.

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 $\sigma$ algebra or not.

Ques 19.

Let A be any set in R , show that $m^*(A) = m^*(A+x)$ where $m^*$ denotes the outer measure.

Ques 20.

Find the measure of the following sets.

i) $E = \bigcap_{n=1}^{\infty }\left ( a-\frac{1}{n}, b \right )$
ii) $E = Q \cup \left \{ 1 ,2 ,3,4 \right \}$
iii) $E = ]5,7[ \cup [7,7.5]$ .

Ques 21.

Show that if f is measurable, then the function $f^a(x)$ given by

$f^a(x) = \begin{Bmatrix} a &if \: f(x)>a \\ f(x) & if \: f(x) \leq a \end{Bmatrix}$
is also measurable.

Ques 22.

Verify Bounded Convergence Theorem for the sequence of functions $\left \{ f_n \right \}$ where

$f_n (x) = \frac{1}{\left ( 1 + \frac{x}{n} \right )^n } , 0\leq x \leq 1, n \in N$

Ques 23.

Find the fourier series of the function $f$ defined by

$f(x) = \begin{Bmatrix} -x^2, -\pi <x\leq 0\\ x^2, 0<x < \pi \end{Bmatrix}$

Ques 24.

State whether the following statements are True or False. Justify your answers.
a) The sequence $\left \{ \left ( \frac{1}{n},\frac{1}{n} \right ) : n \in N \right \}$ 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 $\int_{A}f dm = \int_{A}g dm$ for all $A \in M$ , then $f = g$ .