set element a A A a A a A 1 extensional definition, { } A = {1, 2, 3, 4, 5, 6, 7, 8, 9} 9 1, 2, 3, 4, 5, 6, 7, 8, 9..

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1 ,,.,.,.. 1, 2, 3,., 4),, 4, 5),. 4, 6, 7).,, R A B, 8, (a) A, B 9), (b) {a (a, b) R b B }, {b (a, b) R a A } 10, 11, 12) 2. (a). 11, 13, R S {(a, c) (a, b) R, (b, c) S } (c) R S 14), 1, 15, (d) S R 16) 2. (d). f : X Y. 1, 6, 9,, (e) Y 17) 3, 4, 8, 10, 11, 12, 18, 19), (f) X f (X)., codomain range, (g) Y codomain 7, 9, 14, 16, 19, 20, 21) 3, 4, 8, 9, 10, 11, 12,, (h) f (X) range 15)., (e) (g), (f) (h)., (e) (g) ,.,, ,, 1-2, c /(14)

2 set element a A A a A a A 1 extensional definition, { } A = {1, 2, 3, 4, 5, 6, 7, 8, 9} 9 1, 2, 3, 4, 5, 6, 7, 8, 9... A A = {1, 2,..., 9} {1, 2, 3,...} intensional definition. : { }. x predicate P(x) {x P(x)} A = {2, 4, 6, 8, 10} A = {x x 2 10 } A = {2x x 1 5 } finite set infinite 1-5 A A #A 0 empty set {} 3 A B A B subset A B B A A B A B A B A B A = B A B A B A B A B A B proper subset A B A : A A. : A B B C A C. (1 1) (1 2) 4 family F S F 14, 19) c /(14)

3 A F A A F a A F a F order 5 N : N = {0, 1, 2,...} Z : Q : R : C : A, B 2 1 A, B {x x A x B} A B union A B A, B, C : A = A = A. (1 3) : A A = A. (1 4) : A B = B A. (1 5) : (A B) C = A (B C). (1 6) A 1,, A n A 1 A n I i A i I i A i A i I index set i I 2 A, B {x x A x B} A B intersection A B A, B, C : A = A =. (1 7) : A A = A. (1 8) : A B = B A. (1 9) : (A B) C = A (B C). (1 10) A 1,, A n A 1 A n c /(14)

4 I A i i I A B = A B disjoint A,B A B A + B A B direct sum A, B, C A (B C) = (A B) (A C). A (B C) = (A B) (A C). (1 11) (1 12) A (A B) = A. A (A B) = A. (1 13) (1 14) 3 A B {x x A x B} A B difference A B A B A\B 4 A B (A B) (B A) P Q symmetric difference P Q A B A B A U U A U A complement U U universal set U U A A A A A c U A U = U, U =. (1 15) : A A = U, A A =. (1 16) : A = A. (1 17) De Morgan's laws A, B A B = A B. A B = A B. (1 18) (1 19) U c /(14)

5 A A power set 2 A P(A) 2 A = {S S A} A = {1, 2, 3} 2 A = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} 2 A, A 2 A 2 A = 2 A. (1 20) c /(14)

6 A, B a A, b B (a, b) ordered pair A B {(a, b) a A b B} A B direct product Cartesian product A B A A = A =. (1 21) A, B A B = A B. (1 22) A,B A B R A B 2 binary relation 2 A R domain B codomain A R A A A (a, b) R R(a, b) arb R A B {(b, a) (a, b) R} B A R inverse relation R 1 R, S R = S A 2 1 a A (a, a) R R reflexive 2 (a, b) R (b, a) R R symmetric 3 (a, b) R a b (b, a) R R antisymmetric 4 (a, b) R, (b, c) R (a, c) R R transitive A B R A B B C S B C R S c /(14)

7 {(a, c) R(a, b) S (b, c)} A C R S composition S R R A B S B C T C D T (S R) = (T S ) R. (1 23) n 2 R n R n n = 1 R 1 = R n = 0 R 0 = {(a, a) a A} R 0 identity relation R A S R P A P S R S S S R P- P-closure R transitive closure reflexive transitive closure P P- R + R R + = R 1 R 2 R 3 = R = R 0 R 1 R 2 = i=1 i=0 R i R i (1 24) (1 25) equivalence relation a A, {b A a b} a equivalence class [a] a [a] representative A A F A partition S F S = A S, T F S T S T = A a, b A [a] = [b] [a] [b] = I A F = {[i] i I} A A = [i] F A quotient set A/ i I A F R = {(a, b) S F a S b S } partial order A A partially ordered set, poset, A, ) ( 14, 15, 17) c /(14)

8 1 (A, ) a, b A a b a b a b (A, ) Hasse diagram 1. A a b a b 2. a b a, b a c b c a, b 1 1 (2 {1,2,3}, ) {1, 2} {1} {1, 2, 3} {1, 3} {2} {2, 3} {3} a, b a b b a a b comparable incomparable A S S S chain S S S antichain 3 B A a 0 B a 0 a a B a 0 B maximal B a a 0 a B a 0 B minimal B a 0 B a 0 B a 0 a a B a 0 B maximum B a 0 B a a 0 a B a 0 B minimum B (A, ) total order linear order (A, ) totally ordered set (A, ) A quasi-order c /(14)

9 X Y X Y mapping function f : X Y X domain Y codomain X = dom( f ), Y = codom( f ) f, g dom( f ) = dom(g), codom( f ) = codom(g), x dom( f ) f (x) = g(x) f = g 1 f A dom( f ) f A restriction f A f f A extension 2 f : X Y x X y Y y x image y = f (x) f : x y A X, f (A) = {y x A y = f (x)} f A A = X f (X) f range( f ) f : X Y B Y X {x x X, f (x) B} B f inverse image preimage f 1 (B) f : X Y, g : Y Z x X g( f (x)) f g composition g f g f X Z g f : X Z f, g, h g f h g, h (g f ) = (h g) f, f : X Y x 1, x 2 X x 1 x 2 f (x 1 ) f (x 2 ) f injection 1 1 one-to-one f, g g f f, g g f f g 2 f : X Y y Y f (x) = y x X f (X) = Y f surjection onto f, g g f f, g g f g f c /(14)

10 3 bijection f, g g f f, g g f f g X f : X X X permutation f : X Y y Y y = f (x) x X y x f inverse mapping f 1 f 1 ( f 1 ) 1 ( f 1 ) 1 = f x dom( f ) a codom( f ) f (dom( f )) = {a} f constant function f : X Y x X f (x) = x f identity function c /(14)

11 P(n) n mathematical induction n N P(n) 1 n N P(n) 1. P(0) 2. n P(n) P(n + 1) P(n) 2 (1) n N P(n) 1. P(0) 2. k n P(k) P(n + 1) 3 m, n N P(m, n) 1. P(0, 0) 2. m, n P(m, n) P(m + 1, n) P(m, n + 1) Giuseppe Peano, N Peano axioms c /(14)

12 N 1. N 0 0 N 2. σ : N N σ(n) n (a) σ m n σ(m) σ(n) (b) σ(n) = 0 n N 0 (c) N S 0 n S σ(n) S S = N 2(c) n S 2(c) N recursive definition S, a S S 0 S (S 0 ) b S S c S inductive definition c /(14)

13 , 19) 1 A n {1, 2,..., n} A 2, 4, 9) 2 A A A A N N {0} f (n) = n + 1 N (1), N A B A B equipotent A B cardinality A n {1, 2,..., n} A = n N ℵ 0 ℵ 0 countable set countably infinite set at most countable set uncountable set R 1), Information & computing 61,, ) C.L. Liu,,,,, ),,, Information&Computing 21,, ),, E13,, J. W. R. Dedekind, , 20, 21) c /(14)

14 5),,, ),,, ) R. Garnier and J. Taylor, Discrete Mathematics for New Technology 2nd edition, Taylor & Francis, ),,,, ),,,,,, ) R. Johnsonbaugh, Discrete Mathematics The Jk Computer Science and Mathematics Series, 5th edition, Prentice Hall College Div., ),,, 15,, ), Information Science & Engineering F1,, ),,,, ) J.L. Hein, Discrete Mathematics 2nd edition, Jones & Bartlett Publishers, ) J.K. Truss, Discrete Mathematics for Computer Scientists International Computer Science Series, 2nd edition, Addison-Wesley, ) D.E. Ensley and J.W. Crawley, Discrete Mathematics, Textbook and Student Solutions Manual: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games, Wiley, ) R.J. McEliece, R.B. Ash, and C. Ash, Introduction to Discrete Mathematics, International Edition, ),,, ),,,, 1,, ) R.P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction 4th edition, Addison-Wesley, ) K.A. Ross and C.R. Wright, Discrete Mathematics 5th edition, Prentice Hall, c /(14)

1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,

2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)