1.A Sets, Relations, Graphs, and Functions 1.A.1 Set a collection of objects(element) Let A be a set and a be an elements in A, then we write a A.

Transcription

1 1.A Sets, Relations, Graphs, and Functions 1.A.1 Set a collection of objects(element) Let A be a set and a be an elements in A, then we write a A. How to specify sets 1. to enumerate all of the elements 원소나열법 2. to state the properties that characterizes the elements. 조건제시법 A = {x p(x)} p(x) is a predicate p(x) is either true or false depending on x A = {x U p(x)} A U, U is the universe of discourse x U U is the type of x in A U x in C, Java p(x) attribute of x 3. automata, grammars, programs 8/28/17 Kwang-Moo Choe 1

2 Three cases for two sets A and B 1. subset A B or B A A - B = or B - A = A B = or B A = 2. disjoint A B = 3. in general(incomparable, neither subset nor disjoint) not(a B or B A) and not (A B = ) A \ B and B \ A and A B. A B and A B and A B. A B Venn diagram Truth table 2 n regions 2 n raws 8/28/17 Kwang-Moo Choe 2

3 1.A.2 Binary relation Cartecian product of two sets, A and B A B = {(a, b) a A, b B}. (a, b) A B: an ordered pair. A B = A B. Binary relation R from the set A(domain) to the set B(range). R A B. a A, b B, (a, b) R or a R b. R A B. Inverse of a relation R, R 1 = {(b, a) B A (a, b) R} Composition(Product) of two relations R and S where R A B and S B C. R S = {(a, c) (a, b) R, (b, c) S} Binary relation R on A R A A. Identity relation R on A id A = {(a, a) a A} R A A, R id A = id A R = R. 8/28/17 Kwang-Moo Choe 3

4 Repeated composition(product) of a binary relation R on A. Let R A A. We define R 2 = R R, R 3 = R R R, R n = R R R, iterative deinition for repeated product of binary relations R = R 1. Then we can define R n R m = R n+m, for ( n, m N), n, m 1. R 0 =? If we define R 0 = id A,. Then we can extend the definition R n R m = R n+m, for n, m 0. Another (recursive) definition for repeated product of binary relations R 0 = B id A. basis R n = R R R n-1, n 1. recursion ex) R 3 = R R R 2 = R R R R 1 = R R R R R 0 = B R R R id A = R R R 8/28/17 Kwang-Moo Choe 4

5 1.A.3 A directed graph G = (V, E) is V: a set of vertices, E V V: a set of edges, E: a binary relation on V Some properties of the binary relations 1) R is reflexive, if a A, a R a. id A R R is irreflexive, if a A, a \R a. 2) R is symmetric(=), if a R b impiles b R a. R = R 1 R id A = R is asymmetric(<), if a R b implies b \R a. R R 1 = R is antisymmetric( ), if a R b and a b implies b \R a. R R 1 id A R is asymmetric R is irreflexive. R is asymmetric R is antisymmetric. 3) R is transitive, if a R b and b R c implies a R c. R R R 8/28/17 Kwang-Moo Choe 5

6 Let P = {reflexive, symmetric, transitive}. Then R be P-closure of R, if i) R is P. ii) R R. iii) R is the smallest set among satisfying i) and ii). R satisfying i) and ii), R R. reflexive closure of R, R = R id A. symmetric closure of R, R = R R 1. transitive closure of R, R + = R 1 R 2 R 3 = i N 1 R i where N 1 = {1, 2, 3, }. reflexive-transitive closure of R, R * = R 0 R 1 R 2 R 3 = i N 0 R i where N 0 = {0, 1, 2, }. What is the reflexive (, symmetric) and transitive closure of R in a graph (A, R)? 8/28/17 Kwang-Moo Choe 6

7 Partition of a set A. Let A be a set and A 1, A 2,, A n A. Then we define partition of a, as Par(A) = {A 1, A 2,, A n } is called a partition of A, written Par(A), if 1) i {1, 2,, n} A i = A exhaustive 2) 1 i j n: A i A j =. (pairwise) disjoint Par(A) is the size of the partioion. Ex. Consider A = {a 1, a 2,, a n }. What are Par(A) s? Cover of a set A. Cover(A) = {A 1, A 2,, A n } where if i {1, 2,, n} A i = A exhaustive Power set of a set A, 2 A = P(A) = {B B A} B A B 2 A. 2 A = 2 A. par(a) 2 A. 8/28/17 Kwang-Moo Choe 7

8 A binary relation R on A is equivalence, if R is reflexive, symmetric, and transitive. Par(A) partition of A A binary relation R on A is ((ir)reflexive) partial order, if R is (ir)reflexive, antisymmetric, and transitive. A: partially-ordered set(poset) Let R A A be an equivalence, [a] R = {b A a R b} equivalence class, {[a] R a A} equivalence partition. a set of equivalence classes. a A [a] R = A, exhaustive if a R b, [a] R = [b] R. same equivalent class if a \R b. [a] R [b] R =. (pairwise) disjoint 8/28/17 Kwang-Moo Choe 8

9 Let be a partial order(relation) on A. A A Then (A, ) is called as partially ordered set or poset for short. Let (A, ) be a poset. We define two binary operators on A, ub, lb: A A 2 A ub(a, b) = {c A a c, b c} upper bound( 배수 ) lb(a, b) = {c A c a, c b} lower bound( 약수 ) If a unique lub( ) and a unique glb( ),, : A A A. (A, ) is called as a lattice and a b = min(ub(a, b)) least upper bound( 최소공배수 ) a b = max(lb(a, b)) greatest lower bound( 최대공약수 ) (A,, ) is called an algebra induced by the lattice (A, ). (N, lcm, gcd) is an algebra induced by a lattice (N, ). Boolean algebra, ({f, t},, ), is induced by the lattice ({f, t}, {f t}). Singleton set algebra, (2 {a},, ), is isomorphic to the boolean algebra, ({f, t},, ) with respect to bijection f. 8/28/17 Kwang-Moo Choe 9

10 CS204 TP of Chap. 13(12) boolean algebra p15 Fig. A A -bit string algebra ({0, 1} A,, ) induced by ({0, 1} A, ) is isomorphic to the set algebra (2 A,, ) induced by the lattice (2 A, /28/17 Kwang-Moo Choe 10

11 1.A.4 Algebraic system, semi-group and monoid Let A be a set and be a binary operation on A. : A A A. 1. (A, ) is an algebraic system. if a, b A, a b A closed 2. (A, ) is a semi-group. if a, b, c A, a (b c) = (a b) c a 1 + a a n = + i {1, 2,..., n} a i. binary operation n-ary operation associative indexed set notation 3. (A,, e) is a monoid. if e A. a A, e a = a e = a (A,, e) is called as a monoid. identity 8/28/17 Kwang-Moo Choe 11

12 Let (A,, e) and (B,, ) be two monoids. If i) h: A B is a onto function, A B ii) h(a b) = h(a) h(b), and preserve operation iii) h(e) =. preserve identity Then h is called a homomorphism, and the monoid (B,, ) is called a homomorphic to the monoid (A,, e) w.r.t. h. (A,, e) is called concretization of (B,, ) and (B,, ) is called abstract interpretation of (A,, e). If h is one-to-one and onto, h is called isomorphism. 8/28/17 Kwang-Moo Choe 12

13 1.A.5 A binary relation from A to B is a function from A to B, if 1) a A, (a, b) f, total 2) a A, 1(a, b) f. unique f: A B (a, b) f or a f b or f(a) = b or f a = b. Three faces of a binary relation i) R A B. (a, b) R. i) a set of (ordered) pairs ii) R: A B {false, true}. a R b, iff (a, b) R. ii) a relational operator(, =, ) iii) R: A 2 B. R(a) = {b 1, b 2,, b n }, iff (a, b 1 ), (a, b 2 ),, (a, b n ) R. a A, 1{b 1, b 2,, b n } B or 1{b 1, b 2,, b n } 2 B. R: A 2 B. iii) a set valued function 8/28/17 Kwang-Moo Choe 13

14 Let f: A B. Is f 1 : B A a function? No!!! Function f: A B is onto(surjection; correspondence), if b B, a A.. f(a) = b. A B If f is onto, f 1 is total but not unique function. Function f: A B is one-to-one(injection, 1-1), if a 1, a 2 A, a 1 a 2 implies f(a 1 ) f(a 2 ). if b B.. f(a) = b, 1a A. A B If f is 1-1, f 1 is unique but not total function. Function f: A B is bijective, if f is both 1-1 and onto(1-1 correpondence). b B, if 1a A.. f(a) = b.. A = B If f is 1-1 onto, f 1 is both total and unique, so is a function. 8/28/17 Kwang-Moo Choe 14

15 1.B Set isomorphism and infinite sets If there exists a bijection( 짝짓기, 1-1 onto) f from A to B, two sets A and B have same cardinality, written A = B, and two sets A and B are said to be isomorphic w.r.t. f, written A f B. A set is said to be countable(enumerable), if it has the same cardinality with a subset of N, either finite or infinite and uncountable(uncountably infinite), otherwise. A set is countably infinite, if it has the same cardinality with N. the cardinality of N is denoted as, N =. Let A be countable. Then we can enumerate the set in numeric order. A = {a 0, a 1,, a n } finite for some n 0. A = {a 0, a 1, } countably infinite(enumerable) 8/28/17 Kwang-Moo Choe 15

16 Consider N 1 = {1, 2, 3, }, N 0 = {0, 1, 2, } N 1 = N 0 =,but N 1 N 0. E = {e N e = 2i, i N} E = N =, but E N. I = N {-i i N} I = N =, but N I. Q = N N Q = N =. enumerate (i, j) N N in (natural) numeric order N N = {(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), } f(i, j) = (i+j) + j f(0, 0) = 0 = (i+j)(i+j+1)/2 + j f: N N N is 1-1 and onto Consider f 1 : N N N which is also 1-1 and onto. N N = N =. 8/28/17 Kwang-Moo Choe 16

17 Fact * =. Proof Lexicographical order for *. in the order of size and if same size, in alphabetic order. Let = k. Then we can alphabetic order = {a 1, a 2, a k }, and we can order x * in lexicographical order as follows. * ={( ), (a 1, a 2,, a k ), (a 1 a 1, a 1 a 2,, a 1 a k, a 2 a 1,, a k a k ), (a 1 a 1 a 1, } If x = b 1 b 2 b n *, f: * N is as follows; f(x) = k 0 + k 1 + k n- 1 + b 1 k n-1 + b 2 k n b n k 0 = (k n - 1)/(k-1) + b 1 k n-1 + b 2 k n b n k 0. What is f 1? We enumerate x = a 1 a 2 a n * in numeric order f: * N is one-to-one onto. Q.E.D. 8/28/17 Kwang-Moo Choe 17

18 Consider {0, 1} N : infinite binary strings (See pp.12) and 2 N : power set of natural numbers (Note that 2 A = {B B A} Power set of integers and infinite binary strings are isomorphic. j 2 N b ij = 1 for (b i0, b i1, b ii, ) {0, 1} N where b ij {0, 1}. N 2 N f {0, 1} N. Cantor s diagonal argument Assume {0, 1} N is countable. We can enumerate {0, 1} N, infinite binary string, in numeric order, b 0 = (b 00, b 01,, b 0n, ) b 1 = (b 10, b 11,, b 1n, ) b n = (b 1n, b 1n,, b nn, ) 8/28/17 Kwang-Moo Choe 18

19 Consider b = (b 00, b 11,, b nn, ) (diagonal elements) and b = (b 00, b 11,, b nn, ) where b ii = 0, if b ii = 1; b ii = 1, if b ii = 0. complement of diagonal elements Then b = b i for some i N but b b i for any i N. But b, b {0, 1} N! The assumption {0, 1} N = {b 0, b 1,, b n, } was wrong. Contradiction!!! We fail to enumerate {0, 1} N = {b 0, b 1,, b n, } in numeric order. We conclude that {0, 1} N {b 0, b 1,, b n, } =. {0, 1} N and 2 N are uncountable. Infinite binary strings and power set of integers are uncountable. 8/28/17 Kwang-Moo Choe 19

20 {0, 1} * vs {0, 1} N. {0, 1} * : countably infinite union of finite binary strings = {0, 1} 0 {0, 1} 1 {0, 1} 2 {0, 1} 3 = { } {0, 1} {00, 01, 10, 11} {000,, 111} = {, 0, 1, 00, 01, 10, 11, 000,, 111, } {0, 1} N : uncountaby infinite union of infinite binary strings = { , {} , {0} , {1}, ,, } {0, 1} N = 2 N > {0, 1} * =. {0, 2,, n, } {0, 1, 2, } = N 8/28/17 Kwang-Moo Choe 20

21 Cantor s diagonal argument Complement of diagonal element Russel s paradox S = {x x x} x S, iff x x. But S S, iff S S. contradictory! Halting problem H(P): if halt(p, P) then loop forever elses not halt(p, P) then stop fi What happens if H(H) stops or loops forever? Denial of self recursion * is countable. But is 2 * uncountable. class of languages N. Chomsky strings are countable languages are uncountable 8/28/17 Kwang-Moo Choe 21

22 Finite(countable) Countably infinite natural numbers, integers, rational numbers, finite strings Uncountable Cantor s diagonal argument power set of natural numbers infinite strings real numbers Some informal descriptions on countable and uncountable infiniteness k = k = countable = k = countable But k k = k uncountable(k 2) 8/28/17 Kwang-Moo Choe 22

23 1.C Symbol, vocabulary, string and language Let be a set of symbols ( 문자 ) called as a vocabulary( 어휘 or alphabet) of a language. We define a string x over, whose length is n, is defined as x = a 1 a 2 a n where 1 i n: a i. and we write x = n. A string( 문자열 ) is a sequence of symbols. Example = {a, b,..., z} English alphabet x = school y = boy x = 6 y = 3. Concatenation( ) of two strings x = a 1 a 2 a n and y = b 1 b 2 b m. x y = a 1 a 2 a n b 1 b 2 b m = xy justaxaposed Example x y = xy = schoolboy xy = 9. 8/28/17 Kwang-Moo Choe 23

24 We define concatenation of two strings as : * * * a binary operation on strings (1) x, y *, xy *. closed (2) x, y *, xy yx noncommutative (3) x, y, z *, x(yz) = (xy)z associative (4) We define an empty string denoted as (or ) as an identity element w.r.t. the concatene operation. i.e. x *, x = x = x where = 0. Then ( *,, ) is a noncommutative monoid. We can extend the domain and range of the concatenation from set of strings to the set of languages(set of set of strings) : 2 * 2 * 2 *. Then (2 *,, { }) is a (n induced) noncommutative monoid. 8/28/17 Kwang-Moo Choe 24

25 Four terminologies on the formal language theory(flt). element set symbol( 문자 ) vocabulary( 어휘 ) a, b, c sequence( 열 ) string( 문자열 ) language( 언어 ) x, y, z * L, S, T * Power of an alphabet revisited 0 = B { } basis n = R n-1 for n 1 recursion L, S, T 2 * n = n. * = i N i = { } = 0 i N i = /28/17 Kwang-Moo Choe 25

26 Let B A = {f f: A B}. Then B A = B A. Consider n = {1, 2,, n} = {f f: {1, 2,..., n} } Example x: {1, 2,..., 6} {a, b,..., z} and y: {1, 2, 3} {a, b,..., z} x = school 6 where x(1)=s, x(2)=c,..., x(6)=l; or x = (s, c, h, o, o, l) y = boy = (b, o, y) 3 where y(1) = b, y(2) = o, y(3) = y. strings and functions are isomorphic! + = = i N 1 i where N 1 = {1, 2, 3, }. * = = i N 0 i where N 0 = {0, 1, 2, }. * is a universe of strings including identity( ) ( *,, ) is a (free) monoid over. (unique representation) 2 * is a universe of languages (2 *,, { })is a (free) induced monoid over. 8/28/17 Kwang-Moo Choe 26

27 Reversal, prefix and suffix of strings Let x = a 1 a 2 a n be a string of length n 0 and k 0. x R = a n a n-1 a 2 a 1. reversal of x. recursive definition of reversal of x *. R = B. Let a and x *. Then (ax) R = R x R a. Ex. abc R = R bc R a = R c R ba = R R cba = B cba = cba. Prefix and suffix of a string. For k N, k:x = a 1 a 2 a k, if k n; prefix of x with length k. = x, otherwise. x:k = (k:x R ) R. suffix of x with length k. Ex. 3:school = sch, boy:2 = oy, boy:5 = boy. For k 0: k:, :k: * k = { } 2... k. 8/28/17 Kwang-Moo Choe 27