1.3 Functions and Equivalence Relations 1.4 Languages

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1 CSC4510 AUTOMATA 1.3 Functions and Equivalence Relations 1.4 Languages Functions and Equivalence Relations f : A B means that f is a function from A to B To each element of A, one element of B is assigned Examples: f : defined by the formula f(x) = x A is the domain of the function and B the codomain f and g are equal if and only if they have the same domain and codomain and f (x) = g(x) for every x in the domain Partial functions from A to B may assign values to only some elements of A 2 1

2 Functions and Equivalence Relations (cont d.) The range of a function is the set of elements of the codomain that are actually values of the function { f (x) x A} (a subset of the codomain B) 3 Functions and Equivalence Relations (cont d.) Graph representations of functions that are (or not) one-to-one: One-to-one Not one-to-one Not even a function! 4 2

3 Functions and Equivalence Relations (cont d.) Some functions that are or are not onto their codomains: Onto (but not 1-1) Not Onto (or 1-1) Both 1-1 and onto (bijection) 1-1 but not onto 5 Functions and Equivalence Relations (cont d.) If f : A B is a bijection then we can define the inverse function f 1 from B to A by these two formulas: for every x A and y B, f 1 (f(x)) = x f (f 1 (y)) = y 6 3

4 Functions and Equivalence Relations (cont d.) Relations We can express relationships in several ways: If R is a relation on a set, we can write a is related to b as a R b or as (a, b) R 7 Functions and Equivalence Relations (cont d.) 8 4

5 Functions and Equivalence Relations (cont d.) Theorem: If R is an equivalence relation on A, the equivalence classes with respect to R form a partition of A, and two elements of A are equivalent if and only if they are elements of the same equivalence class. 9 Languages An alphabet is a finite set of symbols usually denoted by Examples: {a,b}, {0,1}, {A, B, C,, Z} A string over is a finite sequence of symbols x : the length of string x n a (x): the number of occurrences of a in string x Null string is a string over any alphabet =

6 Languages (cont d.) The set of all strings over is * Example: {a,b}* = {, a, b, aa, ab, ba, bb, aaa, aab, } A language over is a subset of * Examples: The empty language {, a, aab}, a finite language The palindromes over {a,b}:, a, and baabaab {x {a, b}* n a (x) > n b (x)} {x {a, b}* x 2 and x begins and ends with b} 11 Languages (cont d.) xy is the concatenation of the two strings x and y For every string x, x = x = x xy = x + y Concatenation is associative, i.e., (xy)z = x(yz) If s=tuv then t is a prefix of s, v is a suffix, and u is a substring. Every string is a prefix (and suffix, and substring) of itself. 12 6

7 Languages (cont d.) x, y, z, w, v *, and x=yz, w=yv (1) y is x s prefix. (2) If z, then y is x s proper prefix. (3) z is x s suffix. (4) If y, then z is x s proper suffix. (5) y is the common prefix of x and w. (6) y is the common suffix of x and w. (7) If any common prefix of x and w is y s prefix, then y is the longest common prefix of x and w. (8) If any common suffix of x and w is y s suffix, then y is the longest common suffix of x and w. 13 Languages (cont d.) More conclusions: For any prefix y of x, there is only one corresponding suffix z, such that x=yz; and vice versa. For any proper prefix y of x, there is only one corresponding proper suffix z, such that x=yz; and vice versa. {w w is x s suffix} = {w wisx sprefix}. {w w is x s proper suffix} = {w w is x s proper prefix}. {w w is x s prefix} = {w w is x s proper prefix} {x}, {w w is x s prefix} = {w w is x s proper prefix} +1. {w w is x s suffix} = {w w is x s proper suffix} {x}, {w w is x s suffix} = {w w is x s proper suffix}

8 Languages (cont d.) For languages L 1, L 2 over L 1 L 2, L 1 L 2, and L 1 L 2 are also languages over If L *, the complement of L is a language, * L For languages L 1, L 2 over L 1 L 2 is the language {xy x L 1 and y L 2 } Exponential notation a k = aaa a, where there are k occurrences of a It also applies to strings (x k =xxx x), languages (L k =LLL L), and alphabets ( k = ) a 0 =x 0 =, L 0 ={ } (for every a, x *, L *) 15 Languages (cont d.) If L is a language over, then L* denotes the language of all strings that can be obtained by concatenating zero or more strings in L This operation is known as the Kleene star L*= {L k k } L* for every language L, since L 0 = { } Languages may not be finite, but to use a language we need a finite description L 1 = {ab,bab}* {b} {ba}*{ab}* L 2 = {x {a,b}* n a (x) > n b (x)} 16 8

9 Languages (cont d.) Example: {0, 10}* is the language of strings not containing substring 11 and not ending with 1. What is the language of strings not containing substring 11 and ending with 0? 17 9