1
GATE CSE 2002
Subjective
+5
-0
(a) $$S = \left\{ { < 1,2 > ,\, < 2,1 > } \right\}$$ is binary relation on set $$A = \left\{ {1,2,3} \right\}$$. Is it irreflexive?
Add the minimumnumber of ordered pairs to $$S$$ to make it an $$\,\,\,\,\,$$equivalence relation. Give the modified $$S$$.
Add the minimumnumber of ordered pairs to $$S$$ to make it an $$\,\,\,\,\,$$equivalence relation. Give the modified $$S$$.
(b) Let $$S = \left\{ {a,\,\,b} \right\}\,\,\,\,$$ and let ▢ $$S$$ be the power set of $$S$$. Consider the binary relation $$'\underline \subset $$ (set inclusion)' on ▢ $$S$$. Draw the Hasse diagram corresponding to the lattice (▢$$(S)$$, $$\underline \subset $$)
2
GATE CSE 2002
Subjective
+5
-0
Let $$A$$ be a set of $$n\left( { > 0} \right)$$ elements. Let $${N_r}$$ be the number of binary relations on $$A$$ and Let $${N_r}$$ be the number of functions from $$A$$ to $$A$$.
(a) Give the expression for $${N_r}$$ in terms of $$n$$.
(b) Give the expression for $${N_f}$$ in terms of $$n$$.
(c) Which is larger for all possible $$n, $$ $${N_r}$$ or $${N_f}$$?
(a) Give the expression for $${N_r}$$ in terms of $$n$$.
(b) Give the expression for $${N_f}$$ in terms of $$n$$.
(c) Which is larger for all possible $$n, $$ $${N_r}$$ or $${N_f}$$?
3
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?
4
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.
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