1

GATE CSE 2000

Subjective

+5

-0

A multiset is an unordered collection of elements where elements may repeat ay number of times. The size of a multiset is the number of elements in it counting repetitions.

(a) what is the number of multisets of size 4 that can be constructed from n distinct elements so that at least one element occurs exactly twice?

(b) How many multisets can be constructed from n distinct elements?

2

GATE CSE 2000

Subjective

+5

-0

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$

(a) Prove that $$\left( {0,\,1,\, \otimes } \right)$$ is not a group.

(b) Write $$3$$ distinct groups $$\left( {G,\,\, \otimes } \right)$$ where $$G \subset s$$ and $$G$$ has $$2$$ $$\,\,\,\,\,\,$$elements.

3

GATE CSE 1995

Subjective

+5

-0

Let $${G_1}$$ and $${G_2}$$ be subgroups of a group $$G$$.

(a) Show that $${G_1}\, \cap \,{G_2}$$ is also a subgroup of $$G$$.

(b) $${\rm I}$$s $${G_1}\, \cup \,{G_2}$$ always a subgroup of $$G$$?

(a) Show that $${G_1}\, \cap \,{G_2}$$ is also a subgroup of $$G$$.

(b) $${\rm I}$$s $${G_1}\, \cup \,{G_2}$$ always a subgroup of $$G$$?

4

GATE CSE 1992

Subjective

+5

-0

(a) If G is a group of even order, then

show that there exists an element $$a \ne e$$,

the identifier $$g$$, such that

$${a^2} = e$$

show that there exists an element $$a \ne e$$,

the identifier $$g$$, such that

$${a^2} = e$$

(b) Consider the set of integers $$\left\{ {1,2,3,4,6,8,12,24} \right\}$$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent?

i) Group ii) ring

iii) field iv) lattice

Justify your answer

Questions Asked from Set Theory & Algebra (Marks 5)

Number in Brackets after Paper Indicates No. of Questions

GATE CSE Subjects

Theory of Computation

Operating Systems

Algorithms

Database Management System

Data Structures

Computer Networks

Software Engineering

Compiler Design

Web Technologies

General Aptitude

Discrete Mathematics

Programming Languages