ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 3636.(a)(j)AnC=[2,4]=C-A', (f)A-C=(4,9),
(g) C - A= [-l,2)- (c) A- B=(-a, -31 (d) AA B=(-GO, -31 ~(6, a)
(I) (AA B)AC=(-m, -41 u(-3, l)u(3,6] u [7, a). - (c) D n A' = 0 (i) (D n A) u (D n A') = D.
lO.(b)(-a,-6]u[8,m) (e)(-a,-3)u(-3,3)u(3,a). - (a) [-2, -11 u [I, 21.
- (c) (i) C n E = { f 1 f is the function f(x) = x2 + 3x with domain R)
(iu) A n D = D.
Article 1.3
- (b) Let A = (2, 3, 10) and B = (1, 3, 8, 10). Then (1, 3,4, 5,6,7,8,9, 10) =
(2)' = (A - B)' # A - B' = (3, 10). (g) Let X = {4,7), A = {1,4,7,9),
B = {2,4,6,9). Then X G A, but X u (A n B) = {4,7,9) # {4,9) =
(X u A) n B. - (a) For example, we might try A = {3,4, 5, 8, lo), B = {1,4,5,9), and
X = {2,5,6,7). Then A n X = (5) = B n X, but A n X' = {3,4,8,10) #
(1, 5,9) = B n X'. Or else, we could let A = {6,8,9), B = {2,6,8, 101, and
X = {1,2,9,10). Then A n X' = {6,8) = B n X', but A n X = (91, whereas
B n X = (2, 10). The example seems to indicate that if A and B are distinct
and A n X = B n X, then A n X' can't equal B n X', and vice versa.
(b) An elegant formulation of the idea described informally in (a) is this:
"For any sets A, B, and X, if A n X = B n X and A n X' = B n X', then
A = B." If this conversion is not intuitively evident to you, you will learn
logical principles in Chapters 2 and 3, by which you will be able to make
such conversions yourself systematically. - (b) X n (Y u 2) = (X n Y) u (X n Z) = (-a, -4) u (712, a).
lO.(f)True (j)True.
Article 1.5
- (i) 1 (m) 32 (n) 2 (p) 4.
- (b) 56 three-person committees and 56 five-person committees.
- (b) 4 x 4 x 4 x 4 x 12 = 3072.
- (b) There are 2'' = 1024 possible choices for both A and B, for a combined
total of (1024)2 = 1,048,576. - (a) 3' = 19,683 (b) 36 = 729.
Article 2.1
- (c) a statement (i) a statement only in a specific context (k) not a
statement. - (d) p: the sum of two even integers is even (true), q: the product of two odd
integers is odd (true); p A - q is false. (f) (g) p: April is the name of a month