C TEL SOLUTION
Subject:
Theory of Computation (TOC)
Notes Prepared By:
Ahmad Sir
: 9867597554
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Introduction:
Theoretical Foundations of Computer Sciences (TFCS)
It is also known as Theory of Computation (TOC)
This course is on the Theory of Computation, which tries to answer the
following questions:
• What are the mathematical properties of computer hardware and software?
• What is a computation and what is an algorithm? Can we give rigorous
Mathematical definitions of these notions?
• What are the limitations of computers? Can “everything” be computed?
(As we will see, the answer to this question is “no”.)
Purpose of the Theory of Computation:
“Develop formal mathematical models of computation that reflect real-world computers.”
This field of research was started by mathematicians and logicians in the 1930’s, when they were
trying to understand the meaning of a “computation”. A central question asked was whether all
mathematical problems can be Solved in a systematic way. The research that started in those
days led to computers as we know them today. Nowadays, the Theory of Computation can be
divided into the following three areas:
1. Complexity Theory
2. Computability Theory
3. Automata Theory
Complexity Theory:
The main question asked in this area is “What makes some problems computationally hard and
other problems easy?”
Central Question in Complexity Theory:
Classify problems according to their degree of “difficulty”. Give a rigorous proof that problems
that seem to be “hard” are really “hard”.
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Computability Theory:
The theoretical models that were proposed in order to understand solvable and unsolvable
problems led to the development of real computers. Central Question in Computability Theory:
Classify problems as being solvable or unsolvable.
Automata Theory:
Automata Theory deals with definitions and properties of different types of “computation
models”. Examples of such models are:
1. Finite Automata:
These are used in text processing, compilers, and hardware design.
2. Context-Free Grammars:
These are used to define programming languages and in Artificial Intelligence.
3. Turing Machines:
These form a simple abstract model of a “real” computer, such as your PC at home.
Central Question in Automata Theory :
Do these models have the same power, or can one model solve more problems than the other?
In this course:
In this course, we will study the last two areas in reverse order: We will start with Automata
Theory, followed by Computability Theory. The first area, Complexity Theory, will be covered
in next sem (5th) under DAA (Design analysis and algorithms).
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Syllabus
Theoretical Foundations of Computer Sciences (CSE)
UNIT 1
Mathematical preliminaries –
Sets, operations on sets, relations, strings, closure of relation, Countability and
diagonalization , induction and proof methods- pigeon-hole principle ,concept of
language, formal grammars, Chomsky hierarchy.
UNIT 2
Finite Automaton, regular languages, deterministic & non deterministic finite
automata, €-closures,minimization of automata, equivalence, Moore and Mealy
machine.
UNIT 3
Regular expression, identities, Regular grammar, right linear, left linear, Arden
theorem,Pumping lemma for regular sets, closure & decision properties for
regular sets, Context free languages, parse trees and ambiguity, reduction of
CFGS, Normal forms for CFG .
UNIT 4
Push down Automata (PDA), non-determinism, acceptance by two methods and
their equivalence, conversion of PDA to CFG, CFG to PDAs, closure and decision
properties of CFLs, pumping lemma for CFL
UNIT 5
Turing machines, TM as acceptor, TM as transducers, Variations of TM, linear
bounded automata, TM as computer of function.
UNIT 6
Recursively enumerable (r.e.) set, recursive sets, Decidability and solvability, Post
correspondence Problem (PCP), Introduction to recursive function theory,
primitive recursive functions , Ackerman function
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Subject:
Theoretical Foundations of Computer Science
(TOC)
UNIT –I
Mathematical Preliminaries
Syllabus
1. Sets
2. Operations on sets
3. Relations
4. Closure of Relation
5. Strings
6. Countability and diagonalization
7. Induction and proof methods
8. Pigeon-hole principle
9. Concept of language
10. Formal grammars
11. Chomsky hierarchy
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1. SETS :
Definition:
A set is usually defined as an unordered collection of well defined objects without repetition. The
objects are called elements or member of the set. For examples, the collection of all states in
india , the collection of all students in the college.
The symbols ∈ (read as belongs to) is used to denote the membership in a set. The symbols ∉
(read as does not belongs to) denotes that a particular elements is not a member of a set. Sets are
usually denoted by capital letters.
Let A = {a, b, c, d, e}
Then we can say that a ∈ A, e ∈ A and x ∉ A.
Representation:
There are two different ways of representing of sets.
1. Listing methods:
It is also known as Roaster methods, in this methods all the elements of the set are separated by
commas (,) and the elements are unique.
For example, A = {1, 6, 8, 9}
2. Set builder methods:
It is also known as Ruler methods, in this methods all the elements of the set are denoted by x,
where x satisfies the certain property.
For example, A = {x | x has the property P}
Here the set A denotes all the elements x such that x has the property P.
For example, A = {x | x is the vowel in English} then
A = {a, e, i, o, u}
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Types of sets:
1. Finite set:
A set is said to finite, if it contains finite numbers of elements.
For Example, A = {0, 1, 2, 3}
2. Infinite set:
A set is said to be infinite, if it contains infinite number of elements.
For example, A = {0, 1, 2, 3, …….}
3. Subset:
The subset A is called subset of set B if every elements of set A is presents in set B but reverse is
not true. It is denoted by A ⊆ B.
For example, A = {1, 2, 3} and B = {1, 2, 3, 4, 5} then we can say that A ⊆ B
4. Empty Set:
The set having no element in it is called as empty set. It is denoted by A ={ } and it can be
written as Ø (phi)
5. Power set:
The power set is the set of all the subsets of its elements.
For example, A = {1, 2, 3}
Then no of possible sets in the power set can be calculated as
of |A| = 3 , then the no of possible sets are 8 .
2| A| in the above example value
Q = { Ø, {1}, {2}, {3},{1,2},{1,3},{2,3},{1,2,3} }
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2. OPERATIONS ON SETS :
Usually operations on sets are:
1. Union:
If A and B are two sets then union of A and B denoted as
A ∪ B = {x | x is in A or x is in B}
2. Intersection:
If A and B are two sets then intersection of A and B denoted as
A ∩ B = {x | x is in A and x is in B}
3. Set difference:
If A and B are two sets then difference of A and B denoted as
A - B = {x | x is in A but not in B }
4. Cartesian product:
If A and B are the two sets then the Cartesian product of A and B, denoted as
A×B = {(a ,b) | a is in A and b is in B }
5. Concatenation:
If A and B are two sets then the concatenation of A and B, denoted as
A.B = {ab | a is in A and b is in B}
6. Complement:
Complement of set A is denoted as 𝐴̅ and defined as
𝐴̅ = {𝑥 |𝑥 𝑖𝑠 𝑖𝑛 𝑈 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑖𝑛 𝐴} , where U is a universal set
Example: A = {1, 2, 3,4}, U = {1, 2, 3, 4, 5, 6, ………….}
𝐴̅ = {5, 6, 7, 8, … … … … … … }
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3. RELATIONS :
A set of ordered pairs is known as relations. The first component of the pair is from the set called
as the domain and second component is from the set called range.
If R is a relation and (a,b) is pair in R ,then we write aRb.
Equivalence Relation:
If a relation is Reflexive, Symmetric and Transitive as well as it is said to be Equivalence
Relation
Properties of relations:
If R is a relation on set S , it is said to be
Reflexive:
If aRa exists ∀ ( for all ) a ⊆ S .
Symmetric:
If aRb ⇒ bRa
Transitive:
If aRb and bRc ⇒ aRc
4. CLOSURE OF RELATIONS :
If P is a set of properties of relations R, then the P-closure of relation R is the smallest relation R
that includes all the pair of R and possesses the properties in P.
1. Transitive closure:
Transitive closure of R (R+) is defined as:
i ). If (a, b) ⊆ R then (a, b) ⊆ R+
ii). If (a, b) ⊆ R and (b ,c) ⊆ R then (a, c) is in R+.
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Example : Let, S = { 1, 2, 3}
R = {(1, 2), (2, 2),(2, 3)}
Then, R+ = {(1, 2), (2, 2), (2, 3), (1, 3)}
2. Reflexive closure:
Reflexive closure of R(R*) is defined as:
R* = R+ ∪ {(a,a) | a ⊆ S}
Example: Let, S = {1, 2, 3}
R = {(1, 2), (2, 2), (2, 3)}
Then,
R+ = {(1, 2), (2, 2), (2, 3), (1,3)}
R* = {(1, 2), (2, 2), (2, 3), (1,3)} ∪ {(1,1), (2,2), (3,3)}
R* = {(1, 2), (2, 2), (2, 3), (1,3),(1,1), (3,3)}
Homomorphism :
In group theory, the most important functions between two groups are those that “preserve” the
group operations, and they are called homomorphism. A function f : G → H between two
groups is a homomorphism when
f(xy) = f(x) f(y) for all x and y in G.
Here the multiplication in f(xy) is in G and the multiplication in f(x) f(y) is in H , so a
homomorphism from G to H is a function that transform the operation in G to the operation in H.
5. COUNTABILITY AND DIAGONALIZATION :
Countability:
A set is said to be countable if it is finite or countably infinite .
Infinite sets that can be placed in one- to- one correspondence with the set of natural numbers
(N) said to be countably infinite or countable. Finite sets are always countable. Some infinite sets
are uncountable, eg. Set of real numbers.
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Diagonalization:
The diagonalization principle is based on a simple observation. The diagonalization principle are
used to prove uncountability of sets.
Let A be a finite set, and R be a binary relation on A. We can represent the relation by a square
table, rows and columns representing the elements, and cells having 1 if (a, b) in R other wise 0.
For example, if A = {a, b, c, d}, and R = {(a, b), (a, c), (b, b), (b, d), (c, b), (c, d), (d, a)}, the
table would be:
R
a
b
c
d
a
0
1
1
0
b
0
1
0
1
c
0
1
0
1
d
1
0
0
0
The principle says that the compliment of the diagonal (replacing 1s with 0s and vice versa) is
different from each row.
The reversed diagonal in the example is 1,0,1,1 and you can see that it is different from each
row.
The reversed diagonal differs from the first row in the first element (we have taken the
complement in the diagonal), it differs from the second row in the second element, etc In order to
apply the principle the elements of the set have to be represented as infinite sequences of 0 and 1,
and any infinite sequence of 0 and 1 has to be a representation of some element in the set. Let
us assume that we can order the elements of the set in some way. The binary representation will
result in an infinite table, where each row will correspond to an element in the set.
Let us take now the compliment of the diagonal - it is a representation of some element in the set,
so it should appear somewhere among the rows of the table. However, it differs form the first
row in the first element, it differs from the second row in the second element, etc, and hence it is
not equal to any row in the table. This contradicts the assumption that all elements can be
ordered, and each element corresponds to a row in the table. Hence the set is uncountable.
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Decidability:
If there is an algorithm which terminates after a finite amount of time and correctly decides
whether or not a given number belong to the set is called decidability. A set which is not
computable is called non-computable or undecidable.
6. STRINGS :
Symbol:
It can be defined as the abstract or a user defined entity. It cannot be formally defined ( like
‘Point’ in geometry).
Letter, digits are examples of the symbol.
Alphabet (𝚺):
It can be defined as the some finite collections of symbols are called as the alphabet.
Examples: D = {1, 2, 3, 4, …….,9}
B = {0, 1}
String:
It can be defined as the any finite sequence of symbol over the alphabet is called as the string.
It is generally denoted as the small letters like x, y, z and w etc. the length of string is denoted as
|w|. It is the number of symbol presents in the string.
For example: Σ = {a, b, c}
w = abc
|w| = 3
Prefix:
Any number of leading symbols of the string is known as prefix of a sting.
Example: Let w = abc
Prefix = ∈, a, ab, abc.
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Proper prefix:
The prefix without string itself is called as proper prefix.
Proper prefix: ∈, a, ab.
Suffix:
Any number of trailing symbols of the string is known as suffix of a sting
Example: w = abc
Suffix = ∈, c, bc, abc.
Proper Suffix:
The suffix without string itself is called as proper suffix.
Example: proper suffix = ∈, c, bc.
7. INDUCTION AND PROOF METHODS:
There are two proof methods:
1. Formal proof
2. Inductive proof
1. Formal Proof:
There are various types of proof methods are used to prove the statements as fallow:
I.
Direct proof:
In this proof technique, we prove implication statements that contain two parts
II.
1. An “if –part” which called the premise.
2. An “then-part” which called as conclusion.
Proof by contradiction:
This kind of proof can be applied to all types of statements. Where it is shown that if some
property were true, a logical contradiction occurs, hence the property must be false.
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III.
Proof by contrapositive:
Contrapositive proof technique contains contrapositive of the statements, which is of the form
“ if A then B” concludes “if not A then not B”.
IV.
Proof by deduction:
A deductive proof consists of a sequence of statements whose truth leads some initial statements,
called hypothesis or the given statements, to a concluding statements.
V.
Proof by exhaustion:
Where the conclusion is established by dividing it into a finite number of cases and proving each
one separately.
2. Inductive proof:
Inductive proof based on some observations. It is used to prove recursively defined objects. This
types of proof is also called as proof by mathematical induction.
Principle of Mathematical Induction :
The proof by mathematical induction can be carried out using fallowing steps
1. Basis steps:
In this step we assume the lowest possible value. This is the initial step in the proof of
mathematical induction.
For example: we can prove that the result is true for n=0 or n=1.
2. Induction Hypothesis:
In this step we assign value of n to some other value k. That means we will check whether the
result is true for n = k or not.
3. Inductive step :
In this step, if n = k is true then we check whether the result is true for n = k + 1 or not. If we get
the same result at n = k + 1 then we can state that given proof is true by principle of
mathematical induction.
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Example: prove 12 + 22 + 32 + … … … + 𝑛2 =
n(n+1)(2n+1)
6
Let the given statement be P(n)
12 + 22 + 32 + … … … + 𝑛2 =
1.
n(n+1)(2n+1)
6
Basis step:
For n = 1, we have
P(1): 12 = 1
= (1(1+1)(2*1+1)) / 6
=1
,Thus, P(1) is true
2. Inductive Hypothesis:
Let P (k) be true, where k is a positive integer (k > 1)
12 + 22 + 32 + … … … + 𝑘 2 =
k(k+1)(2k+1)
…………..(1)
6
3. Inductive step :
We will prove that P(k+1) is true
Now,
12 + 22 + 32 + … … … + 𝑘 2 +(k + 1)2
=
=
=
=
=
=
k(k+1)(2k+1)
6
+(k + 1)2
……………..
from (1)
𝑘(𝑘+1)(2𝑘+1)+ 6(𝑘+1)
6
(𝑘+1){2𝑘 2 + 𝑘 +6𝑘+6}
6
(𝑘+1){2𝑘 2 +7𝑘+6}
6
(𝑘+1){ (𝑘+2)(2𝑘+3)}
6
(𝑘+1) (𝑘+2) (2(𝑘+1)+1)
6
thus, P(k +1) is true whenever P(k) is true hence proved.
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8. PIGEON HOLE PRINCIPLE:
If n + 1 or more objects are placed into n boxes, then there is at least one box containing two or
more objects. In other words, if A and B are two sets such that |A| > |B|, then there is no one-toone function from A to B.
If n objects are distributed over m places and n > m, then some place receives at least two
objects.
Example:
Suppose a postman distributes 51 letters (pigeon) in 50 mailboxes (pigeonholes), then it is
evident that some mailbox will contain at least two letters.
Generalized Pigeon Hole Principle:
If n pigeon hole is occupied by Kn+1 pigeons, where K is a positive integers, at least 1 pigeon
hole is occupied by atleast K +1 pigeons.
Example:
Find the minimum number of students in a class so that three of them are born in same month.
Soln: n = 12 (months) which represents the number of pigeon holes
k + 1 =3
k=2
Hence the minimum number of pigeons can be calculated as
= k*n + 1
= 2*12 +1
= 25 students.
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9. CONCEPT OF LANGUAGE :
A Language is defined as the collection of strings over alphabet. It may be finite or infinite
Operations on languages:
Because languages are sets of strings, new languages can be constructed using the set operations.
There are various possible operations are Union, Intersection and difference are also languages
over Σ .besides these operations some more operations can also be performed on languages.
Concatenation:
The concatenation AB of languages A and B is defined by A.B = { (uv) | u ∈ A and v ∈ B }
Example:
Let A = {a, b} B = {aa, ab, ba, bb}
A.B = {aaa, aab, aba, abb, baa, bab, bba, bbb}
Closure operations:
There are two types of closure operations1. Star closure ( kleene*):
Star closure is defined as:
L* = L0 ∪ L1 ∪ L2 ∪ ……….
Example: L = {a, b}
{a,b}* = { ∈} ∪{a, b} ∪{aa, ab, ba, bb}
= {∈, a, b, aa, ab, ba, bb …………..}
L* is the infinite language which contains the all possible string over {a, b}.
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2. Positive closure:
Positive closure is defined as:
L+ = L* - { ∈}
= L1 ∪ L2 ∪ ……….
= {∈, a, b, aa, ab, ba, bb………} – { ϵ}
= { a, b, aa, ab, ba, bb …………..}
L+ is the infinite language which contains the all possible string over {a, b} excluding with ϵ.
10. FORMAL GRAMMARS:
Grammar:
Grammar is the set of rules that generates syntactically correct sentences for the particular
language. Grammar defines the syntax of the language.
The rules of grammar are also called as production rules or syntactical rules.
For example: Dog barks
< Sentence > = < Noun > < verb >
< Noun > = Dog
< Verb > = barks
The sentence “Dog barks” is formed by the rules < Noun > fallowed by < verb >
Components of grammar:
There are two components of grammar
1. Terminal symbols
2. Non-terminal symbols
1. Terminal symbols:
Terminals symbols are those symbols which are the part of generated sentence. In the above
example ‘Dog’ and ‘barks’ are called as terminal symbols since they are the part of generated
sentence “Dog barks”.
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2 .Non-terminal symbols:
Non-terminal symbol are those symbols which take part in the generation or formation of
sentence . They are not the part of generated sentence.
Example:
< Sentence > = < Noun > < verb >
< Noun > = Dog
< Verb > = barks
In above example < Sentence >, < Noun >, < verb > are the non- terminal symbol.
Formal definition:
A phase structure grammar is denoted by 4 tuple of form
G = (V, T, P, S )
Where,
V: finite set of non-terminals
T: finite set of terminals
S: starting symbols
P: production rules of form, 𝛼 → 𝛽
11. CHOMSKY HIERARCHY:
Chomsky himself, in his paper of 1956 and 1959, designated the four types as type 3, type 2,
type 1, and type 0, from most restrictive to most general.
Each level of hierarchy can be characterized by a class of grammar and by a certain type of
abstract machine, or model of computation.
Grammars are classified by the form of their productions.
• Each category represents a class of languages that can be recognized by a different automaton
• The classes are nested, with type 0 being the largest and most general, and type 3 being the
smallest and most restricted.
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This hierarchy is shown in fig below.
• The classifications are:
1. (Type 0) : Unrestricted Grammar :
There are no restrictions on the production of grammar of this type.
This type of grammar permits production of the form, α → β with α ≠ ∈
where , α and β are sentential form i.e. any combination of any number of terminals and nonterminals.
This grammar generates the recursively enumerable languages or every type 0 language forms a
recursively enumerable set. That means, we can construct Turing machine to recognize the
sentences generated by this type of grammar.
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Eg. Let G = (V, T, P, S)
V = (S, A, B, C, D, E)
T = (a, ∈)
S = {S}, S ∈ V
And P is the set of production given by
S → ACaB
Ca → aaC
CB → DB
CB → E
aD → Da
AD → AC
aE → Ea
AE → ∈
As we can see there are no restrictions on the productions, therefore, it is type 0 grammar.
Grammar - Unrestricted Grammar
Language - Recursively enumerable
Automata - Turing machine
2. (Type 1) : Context-Sensitive Grammar :
In this grammar ,every production rule is of the form: α → β where α and β are arbitrary string
of grammar symbol with α ≠ ∈ and | β | ≥ |α| i.e. they can be any string of terminals and nonterminals and the length of the string on the right side must be greater than or equal to the length
of the string on the left side.
As there are restriction that the right hand side must be at least as long as that of the left hand
side.
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Eg. Let the grammar G given as,
G = ( {S, A, B, }, {a, b,c}, P, S)
Where, ‘P’ consists of the production
S → abc | aAbc
Ab → bA
Ac → Bbcc
bB → Bb
aB → aa | aaA
Grammar - Context-Sensitive Grammar
Language - Context-Sensitive
Automata - Linear-bounded automata
3. (Type 2) : Context-Free Grammar :
The context free grammar is defined as
G = (V, T, P, S )
Where,
V: finite set of non-terminals
T: finite set of terminals
S: starting symbol S ∈ V
P: production rules of form, 𝛼 → 𝛽 , where α ∈ V* and β ∈ ( V ∪ T)*
In this grammar, every production rule is of the form:
A → α , where α , is arbitrary string of grammar symbol i.e. the left side of the production rule is
a single non-terminal or variable and the right side can be any string of terminals and nonterminals.
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Example: let the grammar G defined as,
G = ({S}, {a, b}, P, S})
Where, P consists of fallowing productions,
S → aSa | bSb | a | b
Grammar- context-free Grammar
Language -context-free
Automata - Pushdown automaton
4. (Type 3): Regular Grammar :
The regular grammar is defined as
G = (V, T, P, S)
Where,
V: finite set of non-terminals
T: finite set of terminals
S: starting symbol S ∈ V
P: production rules
In this grammar, every production rules is of the form:
A → aB, or A → a, where A and B are non-terminals and ‘a’ is terminals i.e. the left side of the
production rule is a single non terminal fallowed by a non-terminal or only a single terminal.
Example:
Consider grammar:
G = ({S, B, C}, {a, b}, P, S)
Where, P consists of
S → Ca | Bb
C → Bb
B → Ba | b
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Grammar – Regular Grammar
Language - Regular
Automata - Finite-State Automata (FA)
RTMNU QUESTIONS:
1. Write short notes on pigeons hole principal. [S-09]
2. Prove that: [S-10]
P(n) : 1.1! + 2.2! + 3.3! + …….+ n.n!
3. Explain generalized pigeon hole principle. [S-10]
4. Write a short notes on Chomsky hierarchy of languages. Give an example of type 2 grammar
but which is not of type 3. [S-10]
5. Explain the Chomsky hierarchy of language. [S-11, W-11]
6. Prove that by principle of induction
[W-11]
1.2.3 +2.3.4 +3.4.5 +………..+ n(n+1)(n+2) =
𝑛(𝑛 + 1)(𝑛 + 2)(𝑛 + 3)
4
7. Define: Relation, Homomorphism, Countability. [S-12]
8. With the help of mathematical induction prove that : [W-12]
12 + 22 + 32 +…………….+ n2 =
𝑛(𝑛+1)(2𝑛+1)
6
9. What is Diagonalization ? explain with an example. [W-12]
10. Define: Language, Closure and Homomorphism. [S-13]
11. Comment on countability of set of all real number. Use diagonalization. [S-13]
12. Prove the fallowing by principal of induction : [W-13]
(i). 1 + 4 + 7 + …..+ (3n-2) =
𝑛(3𝑛−1)
2
(ii). 2n > n for all n > 1.
All the best…
Notes Prepared by- Ahmad Sir
Contact no: 9867597554
Page 24
13. Discuss the Chomsky hierarchy of language. Identify the type of fallowing grammar: [W-13]
AB → CDB
AB → CdEB
ABcd → abCD |Bcd
B→ b
14. Write short note on countability and diagonalization. [W-13, S-14]
15. Define the following terms with the help of examples: [S-14]
(i). Transitive closure
(ii). Reflexive transitive closure
(iii). Equivalence relation
16. Prove the following by mathematical induction : [S-14]
1 + 2 + 3 + ……..+ (n-1) =
𝑛 (𝑛−1)
2
17. Explain the following with suitable examples: [S-14]
(i). Alphabet set
(ii). Proper prefix
(iii). Proper suffix
(iv). Power set
All the best…
Notes Prepared by- Ahmad Sir
Contact no: 9867597554
Page 25