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Unformatted text preview: A UTOMATA T HEORY S TUDY G UIDE Prepared By Sharafat Ibn Mollah Mosharraf 12 th Batch (0506) Dept. of Computer Science & Engineering University of Dhaka C HAPTER 1 T HE M ETHODS AND THE T ECHNIQUES Concepts 1.1 Easier representation of union and intersection of sets A 1 A 2 A 3 = { x  x ∈ A 1 or x ∈ A 2 or x ∈ A 3 } = { x  x is an element of at least one of the sets A 1 , A 2 and A 3 } More generally, if A 1 , A 2 , … are sets, we can write 1.2 Power set of a set For any set A , the set of all subsets of A is referred to as the power set of A , written as 2 A . The reason for this terminology and this notation is that if A has n elements, then 2 A has 2 n elements. For example, suppose A = {1, 2, 3} . Then, 2 A = { , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} } Notice that and A are both elements of 2 A . Therefore, the empty set () is a subset of every set, and every set is a subset of itself. 1.3 Shortening the description of a set To say that x is an element of the set A , we write x A . Using this notation, we might describe a set A as following: B = { x  x ∈ A and x 10 } which we read “ B is the set of all x such that x belongs to A and x 10”. A common way to shorten this slightly is to write B = { x ∈ A  x 10 } which we read “ B is the set of x in A such that x 10.” 1.4 Logical Quantifiers and Quantified Statements A statement “ there exists an x such that x 2 < 4” is called a quantified statement . The phrase “ there exists ” is called the existential quantifier ; the variable x is said to be bound to the quantifier and is referred to as a bound variable . Existential Quantifier x ( x 2 < 4) “ There exists an x such that x 2 < 4” x ∈ A ( x 2 < 4)  “ There exists an x in A such that x 2 < 4” Universal Quantifier x ( x 2 < 4) “ For every x , x 2 < 4” x ∈ A ( x 2 < 4) “ For every x in A , x 2 < 4” 1.5 Alphabet An alphabet is a finite set of symbols. We denote it by Σ . For example: Σ 1 = { a, b, c, d, …, z } : the set of (lowercase) letters in English Σ 2 = { 0, 1, …, 9 } : the set of (base 10) digits The letters in an alphabet are usually denoted using the letters from the beginning portion in English alphabet, for example, a , b , c , d etc. 1.6 String A string over alphabet Σ is a finite sequence of symbols in Σ . Strings are usually denoted using the letters from the ending portion in English alphabet, for example, s , t , u , v , w , x , y , z etc. The length of a string x over Σ is the number of symbols in the string , and we denote this number by  x  . For example, some of the strings over the alphabet { a , b } are a , baa , aab and aabba ; and we have  a  = 1,  baa  =  aab  = 3 and  aabba  = 5....
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This note was uploaded on 11/08/2011 for the course COMPUTER 102 taught by Professor Asae during the Spring '11 term at Punjab Institute of Computer Science.
 Spring '11
 asae
 Computer Science

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