Chng 2

QUAN H HAI NGI

2.1 NH NGHAa) Tch -cc: Tch -cc ca hai tp A&B l tp: Tch -cc ca cc tp A1, A2, , An l tp:

V d:Cho 2 tp: A = {1; 2; 3}, B = {a, b}AB = {(1; a), (1; b), (2; a), (2; b), (3; a), (3; b)}BA = {(a; 1), (a; 2), (b; 1), (b; 2), (c; 1), (c; 2)}AA = A2 = {(1; 1), (1; 2), (1; 3), (2; 1), (2; 2), (2; 3), (3; 1), (3; 2), (3; 3)}

b) nh ngha: Quan h hai ngi R gia tp A v tp B l tp con ca tch -cc AB. Quan h R trn tp A gi l phn x nu: a A, aRa+ Nu (a, b) R ta vit aRb,+ Nu A = B ta ni R l quan h (hai ngi) trn A.

Quan h R trn tp A gi l i xng nu: a, b A, aRb bRa Quan h R trn tp A gi l phn i xng nu: a, b A, aRb & bRa a = b Quan h R trn tp A gi l bc cu nu: a, b, c A, aRb & bRc aRc

V dXt quan h hai ngi R trn N nh sau: a, b N, aRb (a + b) l s chnHy kim tra cc tnh phn x, i xng, bc cu, phn i xng ca quan h R

2. Quan h ng d mod n: Trn tp s nguyn z, nh ngha quan h nh sau:a, b z, aRb (a b) chia ht cho nV d1. Quan h chia ht: Trn tp N* nh ngha quan h sau:m, n N*, mRn n chia ht cho m

c) Ma trn biu din quan h:Cho 2 tp A = {a1, a2, , an}, B = {b1, b2, , bn}Ma trn biu din quan h gia A&B, k hiu: MR = (mij)mxnSp xp cc phn t ca A&B theo mt trt t no ln lt trn mt hng ngang & hng dc, khi :

V dCho A = {1; 3; 7; 9}, B = {1; 21; 28}Xt quan h hai ngi R gia A&B sau:aRb a l c ca bMt ma trn biu din quan h trn:

2.2 QUAN H TNG NGQuan h R gi quan h tng ng nu n c tnh phn x, i xng v bc cu.V dChng minh quan h ng d mod n l quan h tng nga, b z, aRb (a b) chia ht cho n

HD Tnh phn x: Tnh i xng: R c tnh phn x R c tnh i xng

Bc cu: R c tnh bc cuVy R l mt quan h tng ng

Lp tng ng v phn hoch Cho tp A. Mt phn hoch ca A:S = {A1, A2, , An, /Ai A}Tha cc iu kin sau:

Cho R l mt quan h tng ng trn tp A v xA. Lp tng ng cha x l tp hp cc phn t ca A c quan h vi x, k hiu:V l mt phn hoch ca A. Ghi ch: Tp hp cc lp tng ng S ca A gi l tp thng ca A.

V dCho f(x) = x2 + 2x. Trn tp s thc R, xt quan h tng ng R sau:a, bR, aRb f(a) = f(b)Xc nh cc lp tng ng [0], [1],[2]?[0] = {x/ xR0} = {x/ f(x) = f(0)} = {x/ x2 + 2x = 0} = {0; -2}[1] = {1; -3}, [2] = {2; -4}HD

V dTm cc lp tng ng ca quan h ng d mod 5:a, b z, aRb (a b) chia ht cho 5

Cc lp tng ng:V S = {[0], [1], [2], [3], [4]} l mt phn hoch trn zHD

2.3 QUAN H TH TQuan h R gi quan h th t nu n c tnh phn x, phn i xng v bc cu.V dChng t cc quan h sau l quan h th t:Trn tp s thc R, xt quan h thng thng: a, b R, aRb a b

2. Trn tp N*, xt quan h chia ht sau: a, b N*, aRb b chia ht cho aHD1. Ta kim tra cc tnh cht sau: Tnh i xng: a N*, a a aRa R c tnh phn x

Tnh phn i xng: a, b N*, aRb & bRa a b & b a a = b R c tnh phn i xng Tnh bc cu: a, b, c N*, aRb & bRc a b & b c a c R c tnh bc cuVy R l mt quan h th t

Cho v d v quan h hai ngi: A l tp cc sinh vin trng Vit-Hn, xt mt quan h R trn tp A nh sau:a, bA, aRb a cng khoa vi bSau c th minh ha cc tnh cht ca QH hai ngi.n phn lp tng ng c th minh ha tm cc lp tng ng ca QH R, c 3 lp: KHMT, TMT, THUD v cng l mt phn hoch ca tp A- Cho v d v phn hoch c th cho: Cch chia 1 ming t cho con l 1 phn hoch