Answers and
Solutions to
Selected Exercises
Article 1.1
1.(b) B={-2) (i)={l,2 (m) M={-5,-1,(1&@)/3).- (4 R (4 [-0, $1 (4 C (n) R.
- (a) (i) well defined (ii) not well defined.
- (c) - 16, 32, - 64, 128, - 256.
- (b) for Z, we may say that there exist real numbers a, b, and c such that
a E Z, c E Z, and a < c < b, but b 4 Z. The same characterization may be
used for Q, replacing Z by Q. The statement is clearly true in both cases. - (a) (ii) wj') = (0, {a), {b), {c), {dl, {a, b), {a, c), {a, d), {b, c), {b, d), {c, d),
{a, b, c), (a, b, d), {a, c, d), {b, c, d), S). - If A E A = {Y (Y q! Y), then, by definition of A, we have A q! A, a contradiction.
On the other hand, if A 4 A, then, again, by definition of A, it must not be the
case that A 4 A; that is, A E A, again, a contradiction. The discovery, in 1901,
of this paradox by the British mathematician and philosopher, Bertrand
Russell, had devastating effects in the mathematical world and brought about
essential changes of direction in the developing field of set theory.
Article 1.2
I. (c) A u A' = u (k) A n A = 0 (1) c n c = u.
- (b) (d) (A n C)' = (1,2, 3,4, 5,6,7,8, 10) = A' u C' (k) (n) (A u B) n
C = (9) = (A n C) u (B n C). - (c) (d) (A u C) - (A n C) = (1,2,4,7,8) = A A C. (j) (k) (C - B) n
(C - A) = {2,4,8) = C - (B u A). - (f) A x (B u C) = {(I, 21, (1,3), (L41, (1,5), (1,6), (Lg), (1,9), (1, 1% (7,2),
(7,3), (7,4), (7,5), (7,6), (7,Q (7,9), (7, lo), (9,2), (9,3), (9,4), (9,5), (9,6),
(9,819 (9,9), (9910)).