Set Operations and Properties with Language Examples
Explore set operations like union, intersection, complement, and their properties alongside examples. Learn about the language set, including natural and programming languages. Discover Kleene closure and string concepts with practical examples.
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Lecture2 Operation on set
Basic operation on set 1- Union A B =def { x| x A or x B} 2- Intersection A B =def { x| x A and x B} 3- Difference A B =def { x| x A and x B} 4- Complement A = { x| x A, x U } A = U A
Properties of sets 1. Idempotent Laws (a) X X = X, (b) X X = X 2. Commutative Laws (a) X Y = Y X, (b) X Y = Y X 3. Associative Laws (a) (X Y) Z = X (Y Z), (b) (X Y) Z = X (Y Z) 4. Distributive Laws (a) X (Y Z) = (X Y) (X Z), (b) X (Y Z) = (X Y) (X Z)
5. Identity Laws (a) X = X, (b) X = , (c) X U = U, (d) X U = X 6. Complement Laws (a) X X = U, (b) X X = , (c) (X ) = X (d) X Y = X Y 7. DeMorgan s Laws (a) (X Y) = X Y , (b) (X Y) = X Y
Language: language is the set of all strings of terminal symbols character from alphabet. Types of language 1-Natural language: (e.g.: English, Arabic): It has alphabet:={a, b, c, .z}From these alphabetic we make sentences that belong to the language. 2- Programming language: (e.g.: Cobol, Pascal):It has alphabetic: ={a,b,c,.z , A,B,C,..Z , ?, /, - ,\.} From these alphabetic we make sentences that belong to programming language.
) ( ) ( ) ( Sentences words letters , ) ( entities . ) ( , ) ( grammar Sentences . , Alphabet ( ) , ) ( Set of alphabet . ={a,b,c, ..z} = { 0,1} ={ ......... , , , } grammar
Examples: Let ={x}, be set of alphabet of one letter x. L1={x,xx,xxx,xxxx, } We can write this in an alternate form: L1 = {xn, for n=1,2,3, .} Ex: If a = xx and b = xxx then find a.b , b.a a.b = xxxxx b.a = xxxxx Ex: If a=xx and b=xxx then ab=xxxxx by concatenation a&b Ex: L2={x,xxx,xxxxx, ..} ={xodd} If a = xxx and b = xxxxx are accepted But the catenation of a and b not in L2 Ab = xxxxxxxx Ex: L = { all positive real number } L = { 0.4,0.5,0.1,0.99}
String : : Windows , computer , #1$ , 123 , "mohammed" Length of string string Examples a= windows then length (a) =7 b= xxx then length (b) =3 c= 428 then length (c) =3
Kleene closur Also called (kleene star OR closure) * 0 n 1 n : :+ Examples *= { , x,xx,xxx,xxxx, } += { x,xx,xxx,xxxx, } Ex: If = {0,1}, then *= { , 0,1,00,01,11,111,000, } += { 0,1,00,11,010,000, } Ex : if S = {aa , b}, then S*={ , b,aa,aab,baa,aaaa,aabb,baab,bbaa,bbbb,aaaab,aabaa, }
Example: if = ( the empty set), the * = { }. Ex: If = {a,b,c}, then * = { , a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa...} Example: {"ab","c"}*= { , "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}. Example : {"a", "b", "c"}+= { "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}. Example : {"a", "b", "c"}*= { , "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}.
