This document provides details about an assignment for BCA students in semester 5, including specific questions and a call for submissions to a provided email address. The subjects covered include set theory, recursion, the pigeonhole principle, graph theory, deterministic finite automata, and properties of trees. Instructions for answering questions and evaluation criteria are also included.

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FALL 2014, ASSIGNMENT
PRIGRAM BCA(REVISED 2007)
SEMESTER 5TH
SUBJECT CODE & NAME BC0052 – THEORY OF COMPUTER SCIENCE
CREDIT 4
BK ID B0972
MAX. MARKS 60
Note:Answer all questions.Kindlynote that answersfor 10 marks questionsshouldbe approximately
of 400 words. Each questionis followedbyevaluationscheme.
1. What are the five ways used to describe a set? Describe the set containing all the nonnegative
integers less than or equal to 4.
Answer:A set isa collectionof objects,thingsorsymbolswhichareclearlydefined.
The individual objectsinasetare calledthe membersorelementsof the set.
A setmustbe properlydefinedsothatwe can findoutwhetheranobjectisa memberof the set.
There are twowaysof describing,orspecifyingthe membersof,aset.One wayis by intensional
definition,usingarule or semanticdescription:
A isthe set whose membersare the firstfourpositive integers.
B is the setof colorsof the Frenchflag.
2. What is Recursion Theorem? How do you define n! recursively and compute 5! recursively.
Answer: The Recursion Theorem simply expresses the fact that definitions by recursion are
mathematically valid, in other words, that we are indeed able correctly and successfully to define

2. functions by recursion. Mathematicians implicitly use this fact whenever they define a function by
recursion.
A more general version of the Recursion Theorem would allow the function f to use the argument n as
well as F(n). A still more general version of the Recursion Theorem, called course-of-values recursion,
allows f to use as an argument the entire restriction of the function F∣n to earlier values. (These more
complex versionsof the Recursiontheoremcanbe derivedsolelyfromthe single-value theoremyouhave
stated, by using a function f that takes a
3. State and prove Pigeonhole Principle.
Answer:Inmathematicsandcomputerscience,the pigeonholeprinciple statesthatif nitemsare put into
m pigeonholeswithn>m, thenat leastone pigeonholemust contain more than one item. This theorem
is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a
groupof three gloves".It is an example of a counting argument, and despite seeming intuitive it can be
used to demonstrate possibly unexpected results.
The first formalization of the idea is believed to have been made by Johann Dirichlet in 1834 under the
name Schubfachprinzip("drawerprinciple"or"shelf principle").Forthisreasonitisalso commonly called
Dirichlet's box principle or Dirichlet's drawer principle. In Russian and some other languages, it is
contracted to simply "Dirichlet
4. Prove that in Graph the number of vertices of odd degrees is always even.
Answer: 1. Each edge (including loops) contributes 2 to the vertex order total.
2. This means the vertex order total must be even because it increments by 2 for every edge.
3. For the vertex order total to be even, the number of vertices with odd orders must be even because:
(a) odd number + odd number = even number
(b) odd number + even number = odd number
(c) even number + even number = even number
5. What is Deterministic finite machine? What are the various components of DFA? Illustrate it using
the pictorial representation of DFA.
Answer:Inautomatatheory,a branch of theoretical computer science, a deterministic finite automaton
(DFA)—alsoknown as deterministic finite state machine—is a finite state machine that accepts/rejects
finite strings of symbols and only produces a unique computation (or run) of the automaton for each
input string. 'Deterministic' refers to the uniqueness of the computation.

3. In automatatheory(a branchof computerscience),DFA minimization is the task of transforming a given
deterministicfinite automaton(DFA) intoanequivalentDFA thathas a minimumnumber of states. Here,
two DFAs are called equivalent if they
Q.6 Prove that “A tree G with n vertices has (n–1) edges”
Answer:- We prove this theorem by induction on the number vertices n.
Basic step: If n = 1, then G contains only one vertex and no edge. So the number of edges in
G is n –1 = 1 – 1 = 0.
Inductionhypothesis:The statement is true for all trees with less than ‘n’ vertices. Induction step: Now
let us consider a tree with ‘n’ vertices. Let ‘ek ’ be any edge in T whose end vertices are vi and v j.
Since T is a tree, by there is no other path between vI and v j. So by removing ek from T , we get a
disconnected
graph. Furthermore, T - ek consists of exactly
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Send your semester & Specialization name to our mail id :
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Call us at : 08263069601