1.4 THEOREMS OF SET THEORY 39(b) A U B is the shaded region
Figure 1.9(c) x2 - 3x - 28 > 0 and Ix - $1 5 9
(d) Ix - $1 I and Ix - 81 > $- One way of illustrating the relationship A r B by Venn diagrams would be to
sketch the diagram, as shown in Figure 1.9a, with the understanding that a region
bounded by a "broken" curve is empty and so should never be "shaded in." Thus
we represent A u B in the preceding picture as shown in Figure 1.9b. Using this
convention, illustrate the following theorems by Venn diagrams:
(a) If X r B, then X u (A n B) = (X u A) n B
(b) If A s B, then A u B = B (c) If A c B, then A n B = A
(d) If A G B, then B = A u (B - A) (e) If A c B, then A n B' = 0
lo. Letting U = {1,2, 3,... ,9, lo), answer T or F for each of the following:
(a) (0) r A for every set A (b) 0 E A for every set A
(c) 0 G A for every set A (d) (21 E 9(A) for every set A
(e) 0 r @(A) for every set A *(f) (0) E @(A) for every set A
(9) ((0)) c@(0) (h) (0)u0={0)
0) (0>n0=0 *(i) 05@(W- 0
(k) 9({0)) = {09 {0) 1 (1) {{09 m, {{011}11.4 Theorems of Set Theory
In this article we provide a lengthy list of selected theorems of set theory. The
statement at this point that these results are "theorems," rather than only
conjectures, represents our assertion that the statements are indeed true
and our promise that we will be able to prove each of them once we address
the topic of proof writing in Chapters 4 through 6. We list them at this
stage primarily as a convenient reference for future work, and note that,
until proofs are actually written, we must still regard these statements as
L