Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 14] ORDERED SETS AND LATTICES 365

14.33. See Fig. 14-21.

Fig. 14-21

14.34. See Fig. 14-22.

Fig. 14-22

14.35. See Fig. 14-23.

Fig. 14-23

14.36. Hint: Draw the diagram ofS.

(a) Minimal,e; maximal,a, d.

(b) First,e; Last, none.

(c) {a, d},{b, c}.

14.37. (a) False. Example:N∪{a}where 1/a, andN

ordered by≤. (b) True. (c) True.

14.38. (a) Minimal,a; maximal,dande. (b) First,a; last,

none. (c) Any subset which containscand omitsa;

that is:c,cb,cd,ce,cbd,cbe,cde,cbde. (d)c,cd,

ce,cde. (e)abd,acd,ace.

14.39. (a) Minimal,aandb; maximal,eandf. (b) First,

none; last, none. (c)ace,acf,bce,bcf,bdf.

14.40. (a)Four. (b) None.

14.41. (a) Six. (b) None.

14.42. abcde,abced,acbde,acbed,acebd.

14.43. Eleven.

14.44. a/b,a/c,c/d.

14.45. Minimal, (p, 2) wherepis a prime. Maximal, none.

14.46. (a) an, at, go, or, arm, one, gate, gone, about, occur.

(b) an, about, arm, at, gate, go, gone, occur, one, or.

14.47. (a)‖; (b)>; (c)‖; (d)<.

14.48. 1 c,1y,2a,2c,2z,3b,4b,4z

14.49. (a)e,f,g; (b)a; (c) sup(A)=e; (d) inf(A)=a.