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

22 SET THEORY [CHAP. 1

Fig. 1-10

Fig. 1-11

1.41Use the data to fill in the Venn diagram in Fig. 1-11(b).

Then:

(a) 5; (b) 4; (c) 2; (d) 1; (e) 6; (f) 11; (g) 23; (h) 2.

1.42P (A)has 2^5 =32 elements as follows:

[∅, {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, {1, 4}, {1,

5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {1, 2,

3}, {1, 2, 4}, {1, 2, 5}, {2, 3, 4}, {2, 3, 5}, {3, 4, 5},

{1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 4, 5}, {1, 2, 3, 4},

{1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5},A]

1.43(a) Three elements:[a, b],(c), and {d,e,f}. (b) 3.

(c)P (A)has 2^3 =8 elements as follows:

P (A)={A,[{a, b},{c}],[{a, b},{d,e,f}],

[{c},{d,e,f}],[{a, b}],[{c}],[{d,e,f}],∅}

1.44 LetXbe an element inP (A). For eacha∈A, either

a∈Xora/∈X. Sincen(A)=m, there are 2mdiffer-

ent setsX. That is|P (A)|= 2 m.

1.45 (a) No, (b) no, (c) yes, (d) yes.

1.46 (a) No, (b) no, (c) yes, (d) no.

1.47 (a) No, (b) no, (c) yes.

1.49 [{1,3}, {2,4}, {5,7}, {9}, {6,8}]

1.55 Hint: 7k+^1 − 2 k+^1 = 7 k+^1 − 7 ( 2 k)+ 7 ( 2 k)− 2 k+^1 =

7 ( 7 k− 2 k)+( 7 − 2 ) 2 k

1.58 (a) {1, 2, 3, 7, 8, 9}; (b) {1, 3, 4, 6, 8}; (c) and

(d) {2, 3, 4, 6}.