I
28 SETS Chapter 1
- Given U = (1, 2,3,4), A = (1, 3,4), B = (31, C = (1, 21, find all pairs of disjoint
sets among the six sets A, A', B, B', C, C'. - The solution set to each of the following inequalities can be expressed as the
union of intervals. Find them in each case:
(a) 13x - 231 2 4 (b) 2x2-4x-9620
(c) (x - 5)/(5 - X) < 0 (d) 14x - 171 > 0
(e) (3x2 - 27)/[(x - 3)(x + 3)] > 0 - Solve simultaneously the pairs of inequalities:
*(a) 1x1 2 1 and x2 -450
(b) )4x+8)< 12 and x2+6x+8>0 - Let U be the set of all functions having R as domain and range a subset of R.
Let:
A = { f (f is continuous at each x E R)
B = { f 1 f is differentiable at each x E R)
D = { f I f is a quadratic polynomial)
F = { f 1 f is a linear polynomial)
(a) List all subset relationships between pairs of these six sets.
(6) List all pairs of disjoint sets among these six sets.
(c) Describe, as precisely as possible, the sets:
*(i) CnE (ii) A - B
(iii) D n F *(iv) AnD
(v) A u D (vi) C - E
(vii) F - A (viii) F n A'
(ix) E n F (x) B n E- (a) Consider the Venn diagram displayed in Figure 1.6~. Copy this diagram
- onto a separate piece of paper four times and shade the regions corresponding to
the sets A n B, A' n B, A n B', and A' n B'. [Note: A Venn diagram based on two
sets (i.e., circles) divides the rectangle representing U into four regions, which can be
represented by the preceding four expressions. These sets are mutually disjoint; that
is, the intersection of any two of them is the empty set, and have the totality of U
as their union.]
(6) Draw on a separate piece of paper a Venn diagram based on three sets, that
is, three circles, as in Figure 1.6b. On this figure, shade the regions corresponding
to the sets A n B n C, A n B n C', A' n B' n C, and A' n B' n C'.
(c) In addition to the four regions shaded in (b), how many other nonover-
lapping regions within U are induced by the three circles A, B, and C? Label each
of those remaining regions in a manner consistent with the labeling of the four
regions in (b).
- onto a separate piece of paper four times and shade the regions corresponding to