4

4.1 4.2 4.3 4.4 4.5 4.6

4.1

R R : (empty relation),RA1A2An R : (universal relation)

4-1 n- (n-ary relation) A1A2 An R : A1, A2, , An n- R{(a1, a2, , an) | aiAi} (n - )

4.1 4-2 2- (binary relation) n = 2, RA1A2 , R : A1 A2 (a, b) R aRb , (a, b) R aRb (domain): (range): A (on a set A) : A1A2A A A

4.1 4.1 A{1, 2}, B{3, 4} , A B .[] AB{(1, 3), (1, 4), (2, 3), (2, 4)} 2416 {(1,3)} {(1,4)}{(2,3)} {(2,4)} {(1,3), (1,4)}{(2,3), (2,4)} {(1,3), (2,3)} {(1,4), (2,4)}{(1,3), (2,4)} {(1,4), (2,3)} {(1,3), (1,4), (2,3)}{(1,3), (1,4), (2,4)} {(1,3), (2,3), (2,4)}{(1,4), (2,3), (2,4)}{(1,3), (1,4), (2,3), (2,4)}

4.1 4.2 A{1, 2, 3, 4} R R{(a, b) AA | b a } , R .

[] R{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}

4.1 4.3 , R1{(a, b) | a < b} R2{(a, b) | a>b } R3{(a, b) | ab ab} R4{(a, b) | ab} R5{(a, b) | ab1} R6{(a, b) | ab < 3} (1, 1), (1, 2), (2, 1), (1, 1), (2, 2) .[] (1, 1): R1, R3, R4, R6(1, 2): R1, R6(2, 1): R2, R5, R6(1, 1): R2, R3, R6(2, 2): R1, R3, R4

4.1 4-3 (inverse relation)

R R : R-1 R-1{(b, a)|(a, b)R}

4.1 4.4R{(1, 1), (1, 6), (5, 1), (5, 8), (7, 2)} , R1 .

[]

R1{(1, 1), (6, 1), (1, 5), (8, 5), (2, 7)}

4.2 (1) (arrow diagram)

A, B aA bB aRb a b

4.2

4.5 A{1, 2, 3, 4,} B{a, b, c, d} R{(1, c), (1, d), (2, b), (3, a), (3, c), (4, a)} , R .

4.2 []

4.2 (2) (relation matrix)R : A{a1, a2, , am} B{b1, b2, , bn} . R MR(mij) (, 1im, 1jn)

4.2 4.6 A{1, 2, 3}, B{1, 2} , A B R "aA, bB a>b" R (relation matrix) .[] R{(2, 1), (3, 1), (3, 2)}, R

4.2 4.7 A{a1, a2, a3}, B{b1, b2, b3, b4}, R , R

4.2

[]

R mij1 (ai, bj) R{(a1, b2), (a2, b1), (a2, b3), (a2, b4), (a3, b1), (a3, b3)}

4.2

4.8 A{1, 2, 3} , R{(1, 1), (1, 3), (2, 3), (3, 2)}, S{(1, 1), (1, 3), (2, 1), (2, 2), (3, 3)}. MRS MRS .

4.2 []

4.2 (3) (directed graph)

4-4

(vertex) V, V (edge) E

4.2 4.9 : a, b, c, d : (a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b) ?[]

4.2 4.10 R ? [] R{(1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 3), (4, 1), (4, 3) }

4.2 4.11 A{1, 2, 3, 4, 5, 6, } {R(m, n)AA|mod(m, n)0, m n, , mod n/m } R ?[]

4.3 4-5 (composite relation) A, B, C, R AB, S : BC R S R S{(a, c)AC | (a, b)R (b, c)S, bB }

4.3

4.12 A{2, 3, 8, 9}, B{4, 6, 18}, C{1, 4, 7, 9}. R AB, S : BC,R S .

R{(a, b) | aA, bB, mod(b, a)0}S{(b, c) | bB, cC, bc}

4.3 []

R{(2, 4), (2, 6), (2, 18), (3, 6), (3, 18), (9, 18)}, S{(4, 4), (4, 7), (4, 9), (6, 7), (6, 9)}

R S{(2, 4), (2, 7), (2, 9), (3, 7), (3, 9)}

4.3

4.13 A{1, 2}, B{3, 4}, C{5, 6}, R{(1, 3), (1, 4)}, S{(3, 5), (4, 5)}. R S .

[]

R S{(1, 5)}

4.3 4.14 R S .

4.3

[]

4.3 4-1

A, B, C, D, RAB, SBC, TCD , R (S T)(R S) T .

4.3 [] R (S T)(R S) T (R S) TR (S T) .(1) R (S T)(R S) T (x, y) R (S T) . (x, z) R, (z, y) S T, z B .(z, y)S T, (z, w)S, (w, y)T, wC (x, z)R, (z, w)S (x, w)R S(x, w)R S (w, y)T, (x, y)(R S) T R (S T)(R S) T(2) (R S) TR (S T) .

4.3 4-2

A, B, C , RAB SBC , (R S) 1S1 R1.

4.3 [] (R S) 1S1 R1 S1 R1(R S) 1 .(1) (R S)1 S1 R1 : (y, x)(R S) 1 , (x, y)R S (x, z)R, (z, y)S zB (x, z)R(z, x)R1, (z, y)S (y, z)S1 (y, z)S1, (z, x)R1, (y, x)S1 R1 (R S) 1S1 R1

(2) S1 R1(R S) 1 :

4.3 4.15 {1, 2, 3} R R2 .

4.3 [] R2{(1, 1), (2, 2), (3, 1), (3, 2), (3, 3)}

4.3 4.16 R{(1, 1), (3, 1), (2, 3), (4, 2)} , R2, R3, R4 .

[] R2R R{(1, 1), (2, 1), (4, 3), (3, 1)}R3R2 R{(1, 1), (3, 1), (2, 1), (4, 1)}R4R3 R{(1, 1), (3, 1), (2, 1), (4, 1)}

4.3

4.3 4-3 R A , m, n .

(1) Rm RnR mn(2) Rm RnRn Rm(3) (Rm)nRmn

4.4 4.4.1 (1) (reflexive relation) 4-6 A a, (a, a)R, R (R A )

4.4 4.17 {1, 2, 3, 4} ?

[] (1) , (2) .

4.4 4.18 {1, 2, 3, 4} ?R1{(1, 1), (1, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 1), (4, 4)}R2{(1, 1), (1, 4), (2, 2), (2, 4), (3, 3), (3, 4), (4, 4)}R3{(1, 2), (2, 1), (3, 4), (4, 3), (1, 1), (2, 2), (3, 3)}R4 R5{(1, 1), (2, 2), (3, 3), (4, 4)}R6{(3, 2), (4, 4), (2, 1), (1, 1), (4, 2), (3, 3), (3, 1), (2, 2)}[] R2, R5, R6 : , R1, R3, R4 :

4.4

4.19 "" ?

[]

4.4 (2) 4-7 R A , a, bA (a, b)R (b, a)R , R

4.4

4.20 [ 4.18] {1, 2, 3, 4} ?

[] R3, R4, R5 : ,R1, R2, R6 :

4.4 4.21 . ?

[](1), (2), (3) :

4.4 4.22 ? , Z .(1) R1{(m, n)ZZ : mn}(2) R2{(m, n)ZZ : m n}(3) R3{(m, n)ZZ : mn mn}(4) R4{(m, n)ZZ : nm1}(5) R5{(m, n)ZZ : m < n}(6) R6{(m, n)ZZ : mn 0}[] R1, R3, R6 : R2, R4, R5 : .

4.4 (3) (antisymmetruc relation) 4-8 R A , a, bA , (a, b)R (b, a)R ab

4.4 4.23 ?

4.4

[]

(1) , (2) , . (3) , .

4.4 4.24 1, 2, 3, 4 ?R1{(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 3), (4, 4)}R2{(1, 1), (2, 2), (3, 3), (4, 4)}R3{(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}R4{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}R5{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4),(3, 3), (3, 4), (4, 4)}R6{(3, 4)}

4.4

[]

R4, R5, R6 : R2 : , ( ) R1, R3 :

4.4 (4) (transitive relation) 4-9

R A , a, b, cA, (a, b)R, (b, c)R(a, c)R , R

4.4 4.25 ?

[] (1) (2)

4.4 4.26 ?R1{(1, 2), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}R2{(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (4, 1), (4, 4)}R3{(1, 1), (1, 3), (2, 2), (3, 1), (3, 4), (4, 1), (4, 4)}R4{(1, 1), (1, 3), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 4)}R5{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4)}R6{(1, 1), (1, 3), (3, 1), (3, 4)}R7{(1, 2)}[] R1, R5, R7 : R2, R3, R4, R6 :

4.4

4.27 (directed graph) , , , , (relation matrix) .

4.4

4.4 []

(1) , , .(2) , , .(3) , , .(4) , .

4.4 4-4

R A (1) R IAR(2) R R RR(3) R , IARR R(4) R n RnR ., IA {(a, a) | aA} .

4.4 4-10 (equivalence relation)

A , , .

4.4 4.28 Z R{(m, n)ZZ | mn mn} , R Z .

[](1) nZ, (m, n)R mn (2) (m, n)R, mn nm, mn , nm (n, m)R :

4.4 (3) (m, n)R (n, p)R,

mn, np mpmn, np mpmn, np mpmn, np mp , (m, p)R : R

4.4 4.29 R R{(a,b)RR | ab } , R ?

[]

(1) a a-a = 0 ,aRa :

4.4

(2) aRb ab ba bRa : (3) aRb, bRc ab bc , ac(ab)(bc) aRc :

R

4.4 4-11 (equivalence class)

R A R a : [a] [a]{x (a, x)R}

4.4 4.30 A{1, 2, 3, 4} R .

R{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3), (4, 4)}

[] [1][2]{1, 2}, [3][4]{3, 4}

4.4 4.31 [ 4.28] R .

[]

[a]{a, a}, (a0 )

4.4 4-5

R A , .(1) aRb(2) [a][b](3) [a][b]

4.4 [] (1) (2)aRb , [a][b] : [a][b] [b][a] c[a] aRc R : , aRb bRaR : , bRa , aRc bRc c[b] [a][b]* [b][a]

4.4 (2) (3)[a][b], R : a[a], [a] [a][b][a][a][a]

(3) (1)[a][b] , c[a], c[b] aRc , bRcR cRb aRc, cRb aRb

4.4 4-12 (partition) A A A1, A2, , An .

, Ai A .

4.4 4.32 A{1, 2, 3, 4, 5, 6} A1{1, 2}, A2{3, 4, 6}, A3{5} A ?[]

4.4 4-6

R A A .[] A a a[a] , [a]A aA

[a] [b] [a][b] ( 4-5) {[a] | aA} : A

4.4 4-7 A {A1, A2, , Ak} A R , R A Ai.[]A R : aRb ( a, bAi ) R .(1) aRa , R :

4.4 (2) aRb bRa (a, b Ai ) R : (3) aRb, bRc a, bAi , b, cAjbAiAj, AiAj, a c aRc R :

, , R .

4.5

P : ( ) R : A (R P )S : P R (closure) : R P

4.5 4.33 A{1, 2, 3, 4} R{(1, 2), (2, 1), (2, 2), (3, 2)} , R .

[] S{(1, 1), (1, 2), (2, 1), (2, 2), (3, 2), (3, 3), (4, 4)}

4.5

4-8

R A R R{(a, a) | aA}.

4.5 4.34 A{1, 2, 3} R{(1, 2), (2, 1), (2, 2), (3, 2)} , R .

[] S{(1, 2), (2, 1), (2, 2), (2, 3), (3, 2)}

4.5 4.35 R{(a, b) a>b} .

[] RR1 {(a, b) | a>b} { (b, a) | a>b} {(a, b) | a b}

4.5 4-9

R A R R{(b, a)AA | (a, b)R}RR1.

4.5 4-13 (path)

G a b G (x0, x1), (x1, x2), , (x n1, xn) (, x0a, xnb) . n (; cycle) .

4.5 4.36 , , , ? .

4.5

[]

: 3 : . : 6 , : 1

4.5 4-10

A R a b n (a, b)Rn.

4.5 4-14 (connectivity relation)

A R R* R*{(a, b) | a b }

4.5 4.37 A{1, 2, 3, 4, 5}, R{(1, 1), (1, 2), (2, 3), (3, 4), (3, 5), (4, 5)} .

(1) R2 (2) R*

(1) R2 ={(1,1),(1,2),(1,3),(2,4),(2,5),(3,5)}(2) R* ={(1,1),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}

4.5 4-15 (transitive closure)

R A , R* R .

4.5 4.38 A{1, 2, 3, 4}, R R R* .

4.5

[]

R*{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 4) }

4.5 4.39 {1, 2, 3, 4} R , , , .

4.5 []

(1) .

4.5

(2) : (1, 1), (4, 4) (3) : (3, 1) (4) (R*) :

R*{(1, 1), (1, 3), (1, 4), (2, 2), (3, 3), (4, 1), (4, 3), (4, 4)}

4.6 4-16

A R , , , (A, R) (partially ordered set ; poset) .

4.6 4.40 (Z) .

[] a , a a : a b, b a a=b : a b, b c a c : (Z)

4.6 4.41 (directed graph) .

[] (1), (2), (3)

4.6 4-17 (linear order relation)

(A, ) A a, b a b b a a b (comparable) , A A , .

4.6 4.42 (Z, |) 3 9 ? 5 7 ? , | mod(a, b)0 .

[] 9 | 3 : 9 3 , 5 | 7, 7 | 5 : . | .

4.6 4.43 (Z, ) ?

[] a, b , a b b a ,

4.6 (quasi-order)

a b b c a c ( a b a b, a b)

4.6

: (A, ) : A b a a b b a (cover) : a b A a c b c

4.6 4.44 A = { 1, 2, 3, 4, 5, 6}, n | m mod(m, n) = 0 . (A, |) . , mod(m, n) = 0 m n 0 .

4.6 [] .

4.6 4.45 P({a, b, c}) , (P, ) .[] .

4.6 4.46 .

4.6

[]

(1) : .(2) : (3) : .

4.6 4-11

.

4.6 4-18 (maximal) (minimal)

(P, ) . P x x y y x P , y x y x P .

4.6 (subposet)

P S P , . P .

4.6 4.47 [ 4.44] {1, 2, 3, 4, 5, 6} (1) {2, 3, 4, 5, 6} (2) {1, 2, 3, 6} .[]

4.6 4.48 {a, b, c} P({a, b, c}) . .

4.6 []

4.6 4-19 (largest element;maximum) (smallest element;minimum)

S : (P, ) , : sS, s M M S M = max(S) : sS, m s m S m = min(S)

4.6 [ 4.44] : ( ) : 1

[ 4.45] : {a, b, c}, : { }

4.6 4-20 (upper bound) (least upper bound), (lower bound) (greatest lower bound)

S : (P, )

(1)sS, s x xP, x : P S

4.6 (2) x : P S y : P S , x y x P S x = lub(S)

(3) sS, z s zP, z : P S

(4) z : P S w : P S , w z z P S z = glb(S)

4.6 4.49 [ 4.44] ({1, 2, 3, 4, 5, 6}, |) , , , , .

(1) {2, 3} (2) {4, 6} (3) {3, 6}

4.6 []

(1) : 6, : 6, : 1, : 1(2) : , : 2, 1, glb({4, 6}) = 2(3) : 6, lub({3, 6}) = 6 : 3, 1, glb({3, 6 }) = 3

4.6 4.50 P , , , , , .

(1) {b, c}(2) {d, f}(3) {d, e, f}(4) {b, d, e, f}

4.6

4.6 []

(1) : d, e, g, h. : . : (2) : h. lub({d, f})h. : a, c, : (3) , : h, : a, c, : (4) , : h, , : a

4.6 4-21 (lattice)

(P, ) P x, y {x, y} glb lub , (P, ) . , lub({x, y}) xy, glb({x, y})xy , ( : , *: )

4.6 4.51 P({a, b, c}) , (P, ) .[][ 4.45] ,lub({{a}, {c}}){a}{c}{a, c}lub({{a, b}, {a, c}}){a, b}{a, c}{a, b, c}glb({{a, b}, {c}}){a, b}{c} glb({{a, b}, {b, c}}){a, b}{b, c}{b}

, (P, ) .

4.6 S (P(S), )

lub({A, B, , Z})ABZglb({A, B, , Z})ABZ

4.6 4.52 A{1, 2, 3, 4, 5, 6}, n | m mod(m, n)0 . (A, |) . , mod(m, n)0 m n .

[]{3, 4} .

4.6 4.53 P n | m mod(m, n)0 , (P, |) .[] lub({m, n}) : m, n glb({m, n}) : m, n (), lub({12, 10})60, glb({12, 10})2 ( P, | ) :

4.7 n- 4-25 , , (domain) : D1, D2, , Dn (Cartesian product) : D1 x D2 x x Dn n- (relation) : R , R D1xD2x xDn P171 p177

4.7 n- [] , , , : * (4-)

4.7 n- []

4.7 n- [] :

() : , , , () * () : 1, , CS, 3.83 (n- ) * ,