362 ORDERED SETS AND LATTICES [CHAP. 14
UPPER AND LOWER BOUNDS, SUPREMUM AND INFIMUM
14.49. LetS={a, b, c, d, e, f, g}be ordered as in Fig. 14-11(a). LetA={a, c, d}.(a) Find the set of upper bounds ofA. (c) Does sup(A) exist?
(b) Find the set of lower bounds ofA. (d) Does inf(A) exist?14.50. Repeat Problem 14.49 for subsetB={b, c,e}ofS.
14.51. LetS={ 1 , 2 ,..., 7 , 8 }be ordered as in Fig. 14-11(b). Consider the subsetA={ 3 , 6 , 7 }ofS.
(a) Find the set of upper bounds ofA. (c) Does sup(A) exist?
(b) Find the set of lower bounds ofA. (d) Does inf(A) exist?
14.52. Repeat Problem 14.51 for the subsetB={ 1 , 2 , 4 , 7 }ofS.
14.53. Consider the rational numbersQwith the usual order≤. LetA={x|x∈Qand 5<x^3 < 27 }.(a) IsAbounded above or below?
(b) Does sup(A)or inf(A)exist?14.54. Consider the real numbersRwith the usual order≤. LetA={x|x∈Qand 5<x^3 < 27 }.
(a) IsAbounded above or below? (b) Does sup(A)or inf(A)exist?ISOMORPHIC (SIMILAR) SETS, SIMILARITY MAPPINGS
14.55. Find the number of non-isomorphic posets with three elementsa,b,c, and draw their diagrams.
14.56. Find the number of connected non-isomorphic posets with four elementsa,b,c,d, and draw their diagrams.
14.57. Find the number of similarity mapingsf:S→SwhereSis the ordered set in:
(a) Fig. 14-17(a); (b) Fig. 14-17(b); (c) Fig. 14-17(c).
14.58. Show that the isomorphism relationA∼=Bfor ordered sets is an equivalence relation, that is:
(a) A∼=Afor any ordered setA. (b) IfA∼=B, thenB∼=A. (c) IfA∼=BandB∼=C, thenA∼=C.WELL-ORDERED SETS
14.59. Let the unionSof setsA={a 1 ,a 2 ,a 3 ,...},B={b 1 ,b 2 ,b 3 ,...},C={c 1 ,c 2 ,c 3 ,...}be ordered by:S={A;B;C}={a 1 ,a 2 ,...,b 1 ,b 2 ,...,c 1 ,c 2 ,...}(a) Show thatSis well-ordered.
(b) Find all limit elements ofS.
(c) Show thatSis not isomorphic toN={ 1 , 2 ,...}with the usual order≤.
14.60. LetA={a, b, c}be linearly ordered bya<b<c, and letNhave the usual order≤.(a) Show thatS={A;N}is isomorphic toN.
(b) Show thatS′={N;A}is not isomorphic toN.14.61. SupposeAis a well-ordered set under the relation, and supposeAis also well-ordered under the inverse relation.
DescribeA.
14.62. SupposeAandBare well-ordered isomorphic sets. Show that there is only one similarity mappingf:A→B.
14.63. LetSbe a well-ordered set. For anya∈S, the sets(a)={x|x≺a}is called aninitial segmentofa. Show thatS
cannot be isomorphic to one of itsinitial segments.(Hint: Use Problem 14.21.)
14.64. Supposes(a)ands(b)are distinct initial segments of a well-ordered setS. Show thats(a)ands(b)cannot be
isomorphic. (Hint: Use Problem 14.63.)