360 ORDERED SETS AND LATTICES [CHAP. 14
SupplementaryProblems
ORDERED SETS AND SUBSETS
14.31. LetA={ 1 , 2 , 3 , 4 , 5 , 6 }be ordered as in Fig. 14-17(a).
(a) Find all minimal and maximal elements ofA.
(b) DoesAhave a first or last element?
(c) Find all linearly ordered subsets ofA, each of which contains at least three elements.Fig. 14-1714.32. LetB={a, b, c, d, e, f}be ordered as in Fig. 14-17(b).
(a) Find all minimal and maximal elements ofB.
(b) DoesBhave a first or last element?
(c) List two and find the number of consistent enumerations ofBinto the set{ 1 , 2 , 3 , 4 , 5 , 6 }.
14.33. LetC={ 1 , 2 , 3 , 4 }be ordered as in Fig. 14-17(c). LetL(C)denote the collection of all nonempty linearly ordered
subsets ofCordered by set inclusion. Draw a diagram ofL(C).
14.34. Draw the diagrams of the partitions ofm(see Example 14.4) where: (a)m=4; (b)m=6.
14.35. LetDmdenote the positive divisors ofmordered by divisibility. Draw the Hasse diagrams of:
(a)D 12 ; (b)D 15 ; (c)D 16 ; (d)D 17.
14.36. LetS={a, b, c, d, e, f}be a poset. Suppose there are exactly six pairs of elements where the first immediately
precedes the second as follows:f/a, f/d, e/b, c/f, e/c, b/f(a) Find all minimal and maximal elements ofS.
(b) DoesShave any first or last element?
(c) Find all pairs of elements, if any, which are noncomparable.
14.37. State whether each of the following is true or false and, if it is false, give a counterexample.
(a) If a posetShas only one maximal elementa, thenais a last element.
(b) If a finite posetShas only one maximal elementa, thenais a last element.
(c) If a linearly ordered setShas only one maximal elementa, thenais a last element.
14.38. LetS={a, b, c, d, e}be ordered as in Fig. 14-18(a).(a) Find all minimal and maximal elements ofS.
(b) DoesShave any first or last element?
(c) Find all subsets ofSin whichcis a minimal element.
(d) Find all subsets ofSin whichcis a first element.
(e) List all linearly ordered subsets with three or more elements