Logic, Set Theory and Computability [M. Coppenbarger]
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1 14 Orer (Hnout) Definition 7-11: A reltion is qusi-orering (or preorer) if it is reflexive n trnsitive. A quisi-orering tht is symmetri is n equivlene reltion. A qusi-orering tht is nti-symmetri is n orer (or prtil orer). Definition 14-1: A strit orer on X is n irreflexive, ntisymmetri, trnsitive reltion on X. Definition 14-2: Let (X, ) e n orer n let x,y X. The open intervl from x to y, enote x,y, is efine s x,y = { z X : x z y }. The lose intervl from x to y, enote [x,y], is efine s [x,y] = { z X : x z y }. Definition 14-3: Let (X, ) e n orer. For x,y X, x is overe y y (or y overs x) if x y n x,y =. The overing reltion of X, entoe R (X, ), is the reltion efine y (x,y) R (X, ) if x is overe y y. Definition 14-4: Let (X, ) e n orer. The Hsse igrm of poset (A, ) is the unirete grph representtion r (X, ) so tht if the rrows in the irete eges re inlue, then they woul ll e irete upwr. Definition 14-5: An orer (X, ) is totl if x y or y x for ll x,y X. A strit orer (X, ) is totl if x y or y x for ll x,y X with x y. Definition 14-6: For set X, if there exists n orering suh tht (X, ) is totl, then X is lle totlly orere set. Definition 14-7: Let (X, ) e n orer n X. is miniml element of X if there oes not exist y X suh tht y. is mximl element of X if there oes not exist x X suh tht x, Definition 14-8: Let (X, ) e n orer n X. is minimum element of X, enote min(x), if x for ll x X is mximum element of X, enote mx(x), if x for ll x X. Definition 14-9: Let (X, ) e n orer. We sy tht (X, ) is well-orere if every nonempty suset of X hs minimum element. Definition 14-10: Let (X, ) e n orer. For X, the priniple iel generte y, enote, is efine y = { x X : x }. For X, the wek priniple iel generte y, enote, is efine y = { x X : x }. For S X, the iel generte y S, enote S, is efine y S = { x X : x for ll S }. Definition 14-11: Let (X, ) e n orer. Let X. The priniple filter generte y, enote, is efine y = { x X : x }. Let S X. The filter generte y S, enote S, is efine y S = { x X : x for ll S }. Definition 14-12: Let (X, ) e n orer with S X. If there exists z S suh tht z for ll S, then z is the infimum of S n enote inf S. If there exists z S suh tht z for ll S, then z is the supremum of S n enote sup S. Definition 14-13: An orer (X, ) is lttie if inf{x,y} n sup{x,y} oth exist for every x,y X. Definition 14-14: Let (X, ) e lttie. For x,y X, the meet of x n y, enote x y, is efine s x y = inf{x,y} n the join of x n y, enote x y, is efine s x y = sup{x,y}.
2 Beginning Exerises 14-A-1. Let s e reltion on 4 = { 0, 1, 2, 3 } efine y s = { (0,1), (0,3), (3,1), (3,2) }. Let t e the smllest orer suh tht s t. () Wht re the orere pirs in t? () For (4, t), etermine [0,3], [2,1], n 3,1. () List the orere pirs in R (4, t). () Drw the Hsse igrm of t. (e) Determine ll mximl n miniml elements of (4, t). (f) Evlute for eh 4. (g) Evlute for eh 4. (h) Let S = { 1, 2 }. Evlute inf S n sup S, if they exist. 14-A-2. Let (X, ) e n orer given y the Hsse igrm to the right. () Determine [,]. () Show tht (X, ) is lttie. () Is it possile to n extr orere pir to (X, ) n still hve lttie? f e 14-A-3. The Hsse igrm for the orere set (X, s) where X = {,,, } n the orer s = i X { (,), (,), (,), (,) } is given s the grph in (). Wht is the set n orer for () n ()? () () () e f g f e 14-A-4. Drw Hsse igrms for the following orere sets (X, s). () X = {,,,, e }, s = { (,), (,), (,), (,e), (,e), (,e), (,e) } i X () X = {,,,, e, f, g }, s = { (,g), (,g), (,g), (g,e), (g,f ), (,e), (,f ), (,e), (,f ), (,e), (,f ) } i X () X = {,,, }, s = { (,) } i X () X = 3, s = ( ) For eh exmple isuss whih elements re mximl, miniml, mximum, minimum. 14-A-5. Let X = {,,,, e } n let s = { (,), (,), (,e) }. Prove tht s is strit orer on X, is not totl, n hs three miniml n two mximl elements. 14-A-6. Prove tht if n orere set X hs two istint miniml elements, then it hs no minimum element. 14-A-7. Construt ll orers on the set 3. Whih re totl? 14-A-8. Cn e n orer? A strit orer? 14-A-9. Prove tht s 1 is n orer on X iff s is n orer on X. 14-A-10. Let X e set. Prove tht the reltion s = { (A, B) : A,B X n A B } is n orer on X.
3 14-A-11. Let s e n orer on set X n A X. Prove eh of the following. () Prove tht s (A A) is n orer on A. () Prove tht if s is totl on X, then s (A A) is totl on A. 14-A-12. () If s is n orer on X, prove tht s i X is strit orer. () If s is strit orer on X, prove tht s i X is n orer. 14-A-13. Prove tht n orer n hve t most one miniml element. 14-A-14. Prove tht minimum element is miniml element. 14-A-15. Prove tht if x n y re istint miniml elements, then they re inomprle. 14-A-16. Prove tht [x,y] = x y. 14-A-17. Determine ll ltties with 5 or fewer elements. Construt the Hsse igrms of eh. 14-A-18. Show tht there re 15 istint ltties with 6 elements. Intermeite Exerises 14-B-1. Let X e set n let C X e olletion of susets of X. Prove tht (C, ) is n orer on X. 14-B-2. Let X e n orere set. Let S e finite nonempty totlly orere suset of X n let = sup S. Prove tht = gl S. Give n exmple where S is not totlly orere n the onlusion is flse. 14-B-2. Let S e the set of ll orers on given set X. (S, ) is itself n orere set. () Drw the Hsse igrm of (S, ) in the se where X = 3. () Prove tht t is totl iff t is mximl element of (S, ). 14-B-3. Let s e n ntisymmetri reltion on X. By exmple, prove it my e the se tht there exists no orer t suh tht s t. In other wors, not every ntisymmetri reltion my e extene to n orer. 14-B-4. Let X e n orere set suh tht every nonempty suset A of X with n upper oun in X hs lest upper oun in X. Prove tht every nonempty suset B of X with lower oun in X hs gretest lower oun in X. 14-B-5. Let X n Y e orere sets n let f : X 9 Y e surjetion. Let x < y iff f (x) < f (y) for ll x,y in X. Prove tht f is n injetion. 14-B-6. Let (s i ) i I e nonempty fmily of equivlene reltions on X totlly orere y inlusion. Prove tht «i I s i is n equivlene reltion on X. 14-B-7. Let (s i ) i I e nonempty fmily of orers on X totlly orere y inlusion. Prove tht «i I s i is n orer on X. 14-B-9. Let (X, ) e n orer n let C = { : X } e the olletion of priniple iels of (X, ). Prove eh of the following. () «C X () If X, then»c =.
4 14-B-10. Let (X, ) e n orer n let C = { * : X } = s*(x) e the olletion of wek priniple iels of (X, ). Prove eh of the following. () «C = X () If X, then min( X), if min( X) exists;»c =, otherwise. 14-B-11. Let (X, ) e n orer. Prove if, X n * = *, then =. 14-B-12. Let (X, ) e n orer n let C = { : X } e the olletion of priniple iels of (X, ). On C efine reltion r s r = { (, ) : }. () Prove tht r is n orer on C. () Assign the symol to this orer so tht if (, ) r. Prove tht (C, ) is totlly orere iff (X, ) is totlly orere. 14-B-13. Let R e the set of ll orers on X n S the set of ll strit orers on X. Show tht S ~ R. 14-B-14. Let (X, ) e well-orering. Prove tht (X, ) is totl. Hrer Exerises 14-C-1. Let r e qusi-orer on X. Prove tht there exists n equivlene reltion s on X suh tht the reltion t = { (x / s, y / s) : (x,y) r } is n orer on X / s. 14-C-2. Let s e n orer on X n t n orer on Y. Define reltion r on X Y s r = { ( (x 1,y 1 ), (x 2,y 2 ) ) : (x 1,x 2 ) s n (y 1,y 2 ) t }. Prove tht r is n orer on X Y. [This is known s the prout orer.] 14-C-3. Let s e strit orer on X n t e strit orer on Y. Define reltion r on X Y s r = { ( (x 1,y 1 ), (x 2,y 2 ) ) r : (y 1,y 2 ) t or [(x 1,x 2 ) s n y 1 = y 2 ] }. Prove r is strit orer on X Y. [This is known s the orering y lst ifferenes.] 14-C-4. Let X e set. Given the prtitions S n T of X, S is si to e finer thn T iff A S implies A B for some B T. Define reltion r on the set of prtitions of X s r = { (S,T) : S n T re prtitions of X n S is finer thn T }. Prove tht r is n orer. [Instrutor s Note: Herefter, we will enote S T to men (S,T) r.] 14-C-5. Let R e olletion of orers on set X. () Prove tht if R is nonempty, then»r is n orer. () Provie ounterexmple to show tht «R is not lwys n orer. () Provie neessry n suffiient onition for «R to e n orer. 14-C-6. Assume knowlege of the rel numers R. Define reltion s on X = R [0,1] s s = { (f, g) : f, g R [0,1] n f (x) g(x) 0 for ll x [0,1] }. Prove tht s is n orer. Prove or isprove tht s is totl.
5 14-C-7. Let f, g : X Y. () Prove tht there exists n equivlene reltion r on Y suh tht if f r : Y Y / r is the ssoite quotient mp, then f r g = f r f. () Prove if s is ny equivlene reltion on Y suh tht f s f = f s g, then r s n Y / r Y / s. () Conlue tht there exists h : Y / r Y / s suh tht h f r = f s. 14-C-8. Let S n T e prtitions of X. Define the ross prtition of S n T to e S Ö T = { A B : A S n B T } 1. () Prove tht S Ö T S, S Ö T T, n tht S Ö T is the lest fine prtition finer thn oth S n T. Thus we n efine the supremum of S n T s sup{ S, T } = S Ö T. () Let r n s e the equivlene reltion nturlly orresponing respetively to S n T. Prove tht r s orrespons to S Ö T. () Prove there exists prtition R whih stisfies the following three properties: (i) S R, (ii) T R, (iii) if R is ny other prtition stisfying (i) n (ii), then R R. Thus we n efine R s the infimum of S n T n enote it s R = inf{ S, T }.
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