## Monday, April 23, 2018

2014 (May)
MATHEMATICS
(General)
Course: 601
Full Marks: 80
Pass Marks: 32
Time: 3 hours
The figures in the margin indicate full marks for the questions.
GROUP – A
[(a) Abstract Algebra
(b) Elementary Statistics]
(a) Abstract Algebra
(Marks: 45)
1. (a) Define binary composition on a set X. 1
(b) What do you mean by cyclic group? 1
(c) If denotes the cyclic permutation of a set , then what is the length of ? 1
(d) Is it true that (alternating group) has no subgroup of order six? 1
2. If is a finite non-void subset of a group, then prove that is a subgroup of if and only if for all. 4
3. For each normal subgroup of a finite group, prove that. 2
Or
If and are two groups, andtheir respective identities, and is a homomorphism of into , then prove that
4. If, and, then show that and are disjoint permutations. 2
5. State and prove Lagrange theorem for finite group. 1+5=6
6. Prove that a group of order is cyclic if and only if it has an element of order . 5
Or
What do you mean by a normal subgroup of a group? Prove that subgroup of a group is normal if and only if for every.
7. If is a homomorphism of onto and , any normal subgroup of and then show that is normal subgroup of containing and . 6
8. Prove that every group is isomorphic to a permutation group. 6
Or
Prove that the set of all automorphisms of a group is a group under the resultant composition.
9. Prove that finite non-zero integral domain is a field. 5
Or
If in a ring with unity, for all then prove that is commutative.
10. If is a ring with unity and has no right ideals except and, then show that is a division ring.    5

(b) Elementary Statistics
(Marks: 35)

11. (a) What is the term used for the total number of possible outcomes of a random experiment? 1
(b) What is the sample space when three coins are tossed? 1
(c) State True or False: , where =mean, =SD and =coefficient of variation. 1
(d) What is the shape of normal probability curve with means . 1
12. What is the probability that at least two out ofpeople have the same birthday? Assume 365 days in a year and that all days are equally likely. 4
Or
Three newspapers A, B and C are published in a city. It is estimated from survey that of the adult population: 20% read A, 16% read B, 14% read C, 8% read both A and B, 5% read both A and C, 4% read both B and C, 2% read all three. Find what percentage read at least one of the papers.
13. The first of two samples has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation, then find standard deviation of the second group.     4
14. The chances of solving a mathematical problem correctly by and are and respectively. If the probability of their making a common errors is and they obtain the same answer, then find the probability that their answer is correct. 5
15. Prove that correlation coefficient is independent of the change of origin and scale. 7
Or
Obtain the equations of two lines of regression for the following data. Also obtain the estimate of for =70:
 X : 65 66 67 67 68 69 70 72 Y : 67 68 65 68 72 72 69 71

16. What do you mean by success and failure in binomial probability distribution? and play a game in which their chances of winning are in the ratio 3 : 2. Find A’s chance of winning at least three games out of five games played. 1+5=6
Or
is normally distributed and mean of is 12 and SD is 4. Find out the probability of

GROUP – B
[(a) Discrete Mathematics
(b) Metric Space]
(a) Discrete Mathematics
(Marks: 45)

1. Answer the following questions: 1x5=5
1. Determine the truth value of the following statement: “If 3 is even, then 7 is odd.”
2. Give one example of chain.
3. How many cells are there in a ‘Karnaugh map’ for variables?
4. What is the greatest lower bound of b and d in the lattice given below?
1
d e
a c
0
1. State True or False: “q is a valid conclusion of the premise p if and only if pq is a tautology.”
2. Answer the following questions: 2
1. Write the following statement in symbolic form using suitable symbols: “It is raining and very cold today but not a holiday” where p : ‘Today is a holiday’
q : ‘It is raining today’
r : ‘It is very cold today’
1. Let denoteis younger than. Expressin natural language. 2
2. Write the rule of conditional proof. 2
3. Write the dual statement of the following: where a, b, c are in a lattice L.     2
4. Write the converse and the contra-positive of the following statement: 3
“If Mr. Lohia is a businessman, then he is rich.”
1. Show that the lattice with the following:
‘Hasse diagram’ is complemented but not a distributive lattice: 2+1=3
1

c

a
b

0
3. Answer any four of the following questions: 4x4=16
1. Verify that the proposition is a tautology.
2. Prove that in any lattice for any
1. Draw the Hasse diagram of the lattice where the set of all positive divisors of 30 is and represents the relation of divisibility defined on.
2. For all, in a Boolean algebra B, prove that, where and represent complements of and respectively in B.
3. Obtain the sum-of-products canonical form of Boolean expression of the following:
1. Check the validity of from the following premises: 5
1. What do you mean by Boolean sub-algebra? If is a Boolean algebra and, then show that is a Boolean sub-algebra of. 5

(b) Metric Space
(Marks: 35)

5. State either True or False: 1x3=3
1. Any non-empty set can be regarded as a metric space by defining suitable metric on it.
2. A is open set if and only if A = int (A).
3. The real line is not a complete metric space.
6. Prove that defined by is a metric on 4
7. Let be the usual metric on the set of reals and. Find the distance of from the point 2 and the diameter of. 1 ½ x2=3
8. Define an open subset of any metric space. Is the subset an open subset of the metric space with usual metric? Give reasons for your answer. Can 1 be a limit point of ? 2+1+2+1=6
9. Prove that every convergent sequence in any metric space is a Cauchy sequence. Is the converse true?    3+1=4.
10. Let and be two metric spaces and. When is said to be – 3
1. Continuous at a point
2. Continuous mapping;
3. Uniformly continuous?
Or
Show that the real function defined on IR by is uniformly continuous.
11. Let and be metric spaces and be a continuous mapping. Then prove that is open in whenever is open in . 5
12. Let, then, the metric space with usual metric. Find closure of A, interior of A and derived set of A. Verify whether A is a dense subset of. 2+2+2+1=7
Or

Define a closed set in a metric space. Prove that in any metric space, a subset is closed its complement is open. 2+5=7