1

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.

2

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 $$G$$?

3

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

Discrete Mathematics

Programming Languages

Theory of Computation

Operating Systems

Computer Organization

Database Management System

Data Structures

Computer Networks

Algorithms

Compiler Design

Software Engineering

Web Technologies

General Aptitude