1.AutomataU1.pdf

A symbol (often also called a character) is the smallest building
block, which can be any alphabet, letter, or picture.
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Alphabets are a set of symbols, which are
always finite.
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• A string is a finite sequence of symbols from some alphabet. A
string is generally denoted as w and the length of a string is
denoted as |w|.
• Empty string is the string with zero occurrence of symbols,
represented as ε
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Positive Closure-
L+: It is a Positive Closure that represents a set of all strings
except Null or ε-strings.
Kleene Closure-
L*: It is “Kleene Closure“, that represents the occurrence of
certain alphabets for given language alphabets from zero
to the infinite number of times. In which ε-string is also
included.
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(a) Regular expression for language accepting all combination of g’s
over Σ={g}:
R = g*
R={ε,g,gg,ggg,gggg,ggggg,...}
(b) Regular Expression for language accepting all combination of g’s
over Σ={g} :
R = g+
R={g,gg,ggg,gggg,ggggg,gggggg,...}
Example
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A language is a set of strings with some rules or we can
say- ‘A language is a subset of Σ* ‘.
A language that can be formed over ‘ Σ ‘ can
be Finite or Infinite.
• Finite Language - Countable set of Strings
• Infinite Language - Uncountable set of Strings

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Example of Finite Language
• L1 = { set of string of 2 }
• L1 = { xy, yx, xx, yy }
Example of Infinite Language:
• L1 = { set of all strings starts with 'b’ }
• L1 = { babb, baa, ba, bbb, baab, ....... }

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Automata Set Theory
A set is a collection of distinct elements in which the order of elements
does not matter. If S is the set, then the size of a set is denoted as |S|. If x
is a member in a set S, then it is denoted as x ∈ A and y is a member not
in set S, it is denoted as y ∉ A
A = {1, 2, 3, 4, 5}

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What is Roster Notation?
In roster notation, the elements of a set are represented in a
row surrounded by curly brackets and if the set contains
more than one element then every two elements are
separated by commas.
For example, if A is the set of the first 10 natural numbers so
it can be represented by:
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

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Example:
X = {y: y is a letter in the word dictionary}
We read it as,
“X is the set of all y such that y is a letter in the word dictionary”.

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Types of Sets
1. Singleton Set
2. Finite Set
3. Infinite Set
4. Equal Sets
5. Empty Set
6. Subsets of a Given Set
7. Proper Subset
8. Improper Subset
9. Power Set
10.Universal Set

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(i) Singleton Set
A set consisting of only one element is said to be Singleton
set.
For example :
Set S = {5} , M = {a}
are said to be singleton since they are consists of only one
element 5 and ‘a’ respectively

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(ii) Finite Set –
A set whose number of elements are countable i.e. finite or a
set whose cardinality is a natural number (∈ N) is said to
be Finite set.
For example :
Sets A = {a, b, c, d}, B = {5,7,9,15,78} and
C = { x : x is a multiple of 3, where 0<x<100)
Here A, B and C all three contain a finite number of elements
i.e. 4 in A, 5 in B and 33 in C and therefore will be called finite
sets

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(iii) Infinite Set
A set containing infinite number of elements i.e. whose
cardinality can not found is said to be an Infinite set.
Thus, the set of all natural numbers.
N = {1, 2, 3, 4 . . . .} is an infinite set.
Similarly, the set of all rational numbers between any
two numbers will be infinite. For example,
A = {x : x ∈ Q, 2 < x < 5} is an infinite set.

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(iv) Equal Sets –
When two sets consists of same elements, whether in the
same order, they are said to be equal.In other words, if each
element of the set A is an element of the set B and each
element of B is an element of A, the sets A and B are called
equal, i.e., A = B.
For example,
A = {1,2,3,4,5} and
B = {1,5,2,4,3} ,
then A = B.

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(v) Empty Set –
If a set consists of no element (zero elements), it is said to be the
empty set. It is denoted by ∅. It is also called null set or void set. A
common way of representing the null set is given by
∅ = { x : x ≠ x }, this set is empty, since there is no element which is
not equal to itself. For example, a = a, 2 = 2.

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(vi) Subsets of a Given Set –
Suppose A is a given set. Any set B, each of whose elements is also an
element of A, is called contained in A and is said to be a subset of A.
The symbol ⊆ stands for “is contained in” or “is subset of”. Thus, if “B is
contained in A” or “B is subset of A”, we write
B ⊆ A.
When B is subset of A, we also say ‘A contains B’ or ‘A is superset of B.
The symbol ⊇ is read for “contains” this A ⊇ B means “A contains B”.
Example :
If A = (3, 5, 7), B = (3, 5, 7, 9) than A ⊆ B since every element of A is also an
element of B. But B ⊄ A since 9 ∈ B while 9 ∉ A

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(vii) Proper Subset –
If B is a subset of A and B ≠ A, then B is said to be proper subset of A. In
other words, if each element of B is an element of A and there is at least
one element of A which is not an element of B, then B is said to be a proper
subset of A. “Is proper subset of” is symbolically represented by ⊂.
Also, the empty set ∅ is a proper subset of every set except itself.
Improper Subset –
Set A is called an improper subset of B if and only if A = B.
Note : Every set is an improper subset of itself

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(viii) Power Set –
The set of all subsets of a given set A, is said to be the power set of A.
The power set of A is denoted by P(A).
If the set A= {a, b, c}
then its subsets are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c}.
Therefore, P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c} }.

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(ix) Universal Set –
A set which consists of all the elements of the considering sets is said to be
the Universal set for those sets. It is generally denoted by U or S.
For example :
Consider the following sets,
A = {a, b, c, d, e} ;
B = {x, y, z} and
U = {a, b, c, d, e, f, g, h, w, x, y, z}
Here, U is the universal set for A and B, since U contains all the elements
of A and B.

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Finite Automata(FA) is the simplest machine to recognize patterns. The finite automata
or finite state machine is an abstract machine that has five elements or tuples. It has a
set of states and rules for moving from one state to another but it depends upon the
applied input symbol. Basically, it is an abstract model of a digital computer. The
following figure shows some essential features of general automation
Introduction of Finite Automata

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