RGU Question Paper Mathematics Semester VI 2020: Number Theory and Metric Space

 

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B.Sc (VI Semester) Examination

MATHEMATICS

Paper : MATH–362 (New Course)

( Number Theory and Metric Space )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five of the following questions : 2×5=10

(a) Find $\varphi \left(180\right)$.

(b) What is well-ordering principle? Describe it with an example.

(c) Show that square of any odd integer is of the form $8k+1$.

(d) Define limit and limit point in context of metric space. Are they same?

(e) Prove with an example that a continuous function may not always be a uniformly continuous function.

(f) State the Heine-Borel theorem.

(g) Find interior and closure of the set $(0,1]$ with respect to usual metric on $\mathbb{R}$. Is this an open set?

(h) Find gcd of 180 and 580 using Euclidean algorithm.

2. Answer any four of the following questions : 5×4=20
(a) Prove that—
(i) if $a\equiv b (mod n)$ and $b\equiv c (mod n)$, then $a\equiv c (mod n)$;
(ii) if $a\equiv b (mod n)$, then $a+c\equiv b+c (mod n)$ and $ac\equiv bc (mod n)$;
where $a,b,c$ are integers, and $n$ is a fixed integer.

(b) State and prove division algorithm.

(c) Define Euler’s phi function. If $p$ is a prime and $k>0$, then show that
$\varphi \left(p^k\right)=p^k-p^{k-1}=p^k\left(1-\frac{1}{p}\right)$

(d) Let $d$ be a metric on a non-empty set $X$, and let $d_1(x,y)=\frac{d(x,y)}{1+d(x,y). }$Show
that $d_1$ is also a metric on $X$.

(e) Let $X$ be a metric space and let $F$ be a subset of $X$. Show that $F$ is closed if and only if its complement $F^c$ is open.

(f) Let $X$ be a complete metric space and let $Y$ be a subspace of $X$. Prove that $Y$ is complete if and only if it is closed.

(g) Prove that open sphere is an open set.

3. Answer any five of the following questions : 10×5=50

(a) Define dense and nowhere dense sets. If $\left\{A_n\right\}$ is a sequence of nowhere dense sets in a complete metric space $X$, then prove that there exists a point in $X$ which is not in any of the $A_n$.

(b) (i) Let $X$ be a metric space and let $\left\{x_n\right\}$ and $\left\{y_n\right\}$ be sequences in $\mathrm{X }$such that $x_n\to x$ and $y_n\to y$. Prove that $d(x_n,y_n)\to d(x,y)$.

(ii) Let $X$ and $Y$ be metric spaces and let $f$ be a mapping from $X$ into $Y$. Prove that $f$ is
continuous if and only if $x_n\to \mathrm{x }$in $X$ implies that $f\left(x_n\right)\to f(x)$in $Y$. 5+5=10

(c) Briefly describe the concept of compactness of metric space. What do you mean by Bolzano-Weierstrass property? Prove that a metric space is sequentially compact if and only if it has the Bolzano-Weierstrass property.

(d) (i) Test for uniform continuity of the following function :
$f\left(x\right)=\frac{1}{1-x} \text{on} (0,1)$

(ii) Show that the set of complex numbers with respect to usual metric is complete

(e) Give a complete description of Chinese remainder theorem with proof. Solve the following system of congruences :
$x\equiv 2 (mod 3)$
$x\equiv 3 (mod 5)$
$x\equiv 2 (mod 7)$
using Chinese remainder theorem. 6+4=10

(f) (i) Define the functions $\gamma$ and $\sigma$. Show that $\gamma$ and $\sigma$ are both multiplicative functions.

(ii) Prove that if $f$ is a multiplicative function and
$F\left(n\right)=\sum _{d|n}^{}f\left(d\right)$
then $F$ is also multiplicative.

(g) (i) Give a complete description of Euclidean algorithm. Find $\mathrm{x }$and $y$ satisfying the following equation :
$gcd\left(\mathrm{24,138}\right)=24x+138y$

(ii) Determine all solutions in the integers of the following Diophantine equation :
$56x+72y=40$

(h) (i) State and prove Euler’s theorem.
(ii) Find the values of $\gamma \left(12\right)$ and $\sigma \left(12\right)$. 6+4=10

(i) (i) State and prove Fermat’s theorem.
(ii) Prove that the mapping
$d(x,y)=\left\{\begin{array}{ll}1& if x=y\\ 0& otherwise\end{array}\right.$
is a metric on $X$, where $X$ is a non-empty set. 6+4=10