RGU Question Paper Mathematics Semester VI 2021: Number Theory and Metric Space

 June-2021

B.A/B. Sc. (VI Semester) Examination

MATHEMATICS

Paper: MATH-362 (N/C)

(Number Theory and Metric Space)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 𝟐 × 𝟓 = 𝟏𝟎

(a) Find the GCD of the pair of numbers 360 and 520 using Euclidean algorithm.

(b) Give a brief description of the well-ordering principle with suitable example.

(c) Find the values of 𝜙(520) and 𝜎(180).

(d) Prove or disprove: “image of an open set under a continuous function is an open set in metric space”.

(e) Justify with your logics: “for any 𝑎, 0 < 𝑎 < 1, the interval; [0, 𝑎) is neither a closed nor an open subset of the set of real numbers with respect to the usual metric space”.

(f) Examine the completeness of the set of complex numbers with respect to the usual metric.

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) State and prove Heine-Borel theorem.

(b) In a metric space 𝑋, prove that a subset 𝐹 is closed if and only if its complement is open.

(c) 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 $A_n$’s.

(d) For arbitrary integers 𝑎, 𝑏, 𝑐, the following holds:

(i) if 𝑎|𝑏, 𝑏 ≠ 0, then |a| ≤ |b|,

(ii) If 𝑎|𝑏 and 𝑎|𝑐, then 𝑎|𝑥𝑏 + 𝑦𝑐, for arbitrary integers 𝑥 and 𝑦.

(e) Give a complete description of linear Diophantine equation. Determine all integers solutions of the following linear Diophantine equation 56𝑥 +72𝑦 = 40.

(f) Sate and prove Euler’s theorem.

III. Answer any five parts: 𝟏𝟎 × 𝟓 = 𝟓𝟎

(a) If 𝑎 and 𝑏 are integers (not both of which are zero), then there exists integers 𝑥 and 𝑦 such that gcd(𝑎, 𝑏) = 𝑎𝑥 + 𝑏𝑦. Hence, deduce that if 𝑎 and 𝑏 are relatively prime, then 1 = 𝑎𝑥 + 𝑏𝑦 for some integers 𝑥 and 𝑦. Also, find 𝑥 and 𝑦 such that gcd(18,40) = 18𝑥 + 40𝑦. (5+2+3)

(b) Define continuity and sequential continuity of a function in a metric space with suitable examples. Let 𝑋 and 𝑌 be metric spaces and let 𝑓 be a mapping from 𝑋 to 𝑌. Prove that 𝑓 is continuous at $x_0$ if and only if $x_n\to x_0\Rightarrow f\left(x_n\right)\to f\left(x_0\right)$. (3+5+2)

(c) (i) Let 𝑓 be a continuous real function defined on ℝ which satisfies the functional equation 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) + 𝑓(𝑦). Show that this function must have the form 𝑓(𝑥) = 𝑚𝑥 for some real number 𝑚. (6)

(ii) Give a brief description of sequentially compact metric space with suitable examples. (4)

(d) Define uniform continuity in metric space. Examine, whether or not the following functions are uniformly continuous or not on the open unit interval (0, 1): $f\left(x\right)=\frac{1}{1-x}, f\left(x\right)=x^3$. (2+8)

(e) State and prove fundamental theorem of arithmetic. Given that 𝑝 is a

prime and 𝑝|𝑎, prove that $p^n|a^n$. (6+4)

(f) If gcd(𝑎, 𝑏) = 𝑝, a prime, what are the possible values of $gcd(a^2,b^2)$ , $gcd(a^2,b)$ and $gcd(a^3,b^3)$. Define congruence relation. For any arbitrary integers 𝑎 and 𝑏, prove that 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛) if and only if 𝑎 and 𝑏 give the same non-negative remainder when divided by 𝑛. (5+1+4)

(g) State Chinese Remainder theorem. Solve the following set of simultaneous congruences:

𝑥 ≡ 5(𝑚𝑜𝑑 11), 𝑥 ≡ 14(𝑚𝑜𝑑 29), 𝑥 ≡ 15 (𝑚𝑜𝑑 31).

(h)(i) Solve the linear congruence 𝑥 ≡ 15 (𝑚𝑜𝑑 29) . (5)

(ii) Let (𝑋, 𝑑) be a metric space and let $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}, \forall x,y\in X$. Prove that $d_1$ is also a metric on 𝑋. (5)