RGU Question Paper Mathematics Semester VI 2020: Topology and Boolean Algebra(Old)

 

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BA/B.Sc (VI Semester) Examination

MATHEMATICS

Paper : MATH–363 (Old Course)

( Topology and Boolean Algebra )

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 the following questions (any five) : 2×5=10

(a) Define ‘relation’ on sets with an example.

(b) What are sublattices? Give an example of sublattice.

(c) Define modular lattice. Show that every distributive lattice is modular.

(d) What do you mean by a distributive lattice? Draw the diagram of a lattice which is not distributive.

(e) What do you mean by neighbourhood in a metric space? 

(f) Define Cauchy sequence in a metric space.

(g) What is discrete topology?

(h) If T = { f, X, {b}, {a,b}, {a,b,d}} is a topology on X = {a, b, c,d}, then find the closure of {a}.

2. Answer the following questions (any four ) : 5×4=20

(a) What is poset? Give an example of poset with explanations.

(b) Let L be any lattice. Then show that $a\vee (b\vee c)=(a\vee b)\vee c$ for all $a,b,c\in L$.

(c) Define a homomorphism from a lattice L to a lattice M. Let $f:L\to \mathrm{M }$be defined as $f\left(a_1\right)=f\left(c_1\right)=f\left(d_1\right)=a_{\mathrm{2 }}$and $f\left(b_1\right)=b_2$. Find whether f is a homomorphism or not.

(d) If (X,d) is a metric space, then prove that the function f defined by
$f(x,y)=\frac{d(x,y)}{1+d(x,y)}$

is also a metric on X.

(e) Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces. Prove that the function $f:X\to Y$ is continuous if and only if $f^{-1}\left(v\right)$ is open in X whenever v is open in Y.

(f) If $T_1$ and $T_2$ are two topologies on same set X, then show that $T_1\cap T_{\mathrm{2 }}$is also a topology on X. Also show by an example that $T_1\cup T_2$ is not necessarily a topology on X.

(g) Show that a closed subset of a compact topological space is compact.

3. Answer the following questions (any five) : 10×5=50
(a) (i) Show that in a lattice L, $a\wedge (b\wedge c)=(a\wedge b)\wedge c$ for all $a,b,c\in L$. 4

(ii) In any lattice, state whether the following statements are equivalent : $1. a\wedge (b\wedge c)=(a\wedge b)\vee (a\wedge c) 2. a\vee \left(b\wedge c\right)=\left(a\vee b\right)\wedge \left(a\vee c\right)$

for all $a,b,c\in L$. 3+3=6

(b) (i) Show that every chain is distributive lattice. 5

(ii) Prove that pentagonal lattice is not modular. 5

(c) (i) Show that in a distributive lattice $(L,\le ), a\wedge b=a\wedge c$ and $a\vee b=a\vee c$ imply that b = c. 6

(ii) Write a short note on compound propositions. 4

(d) What is Boolean algebra? Let the operation $(+), (·)$ and $(\text{'})$ in $D_6=\{1,2,3,6\}$ be defined by $a+b=lcm (a,b); a·b=gcd (a,b); a\text{'}=6/a$. Show that $(D_6,+,·,\text{'},1,6)$ is a Boolean algebra.

(e) Prove that in a Boolean algebra B for all $a\in B, \left(i\right) a+a=a, \left(ii\right) a+1=1, \left(iii\right) a·a=a, \left(iv\right) a·0=0$ and (v) $a+(a·b)=a$. 10

(f) (i) Show that there are 2n minimal Boolean functions in n variables.

(ii) Express $xy\text{'}+xz+x\text{'}y$ in disjunctive normal form and $xy\text{'}+xz+x\text{'}y$ in the conjunctive
normal form. 10

(g) Define closed sphere and closed set. Show that a subset F of a metric space (X,d) is closed if and only if complement of F is open. Also show that arbitrary intersection of closed set is closed.

(h) Define a metric space. Show that the function d defined as $d(x,y)=\left|x-y\right|, \forall x,y\in \mathbb{R}$ is a metric on $\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers. Also, show that $\mathbb{R}$ is a complete metric space with respect to the metric d. 10

(i) (i) If X is any infinite set and T be the collection of subsets of X consisting of $\varphi$ and complements of finite subset of X, then show that T is a topology on X. 5

(ii) If $(X,T_1)$ and $(Y,T_2)$ be two topological spaces, then prove that a one-one onto map $f:X\to Y$ is a homomorphism if and only if $f\left(\stackrel{-}{A}\right)=\overline{f\left(A\right)}$ for any
$A\subset X$. 5