38 SETS Chapter 1(e) B u (A - B) = A
(f) IfAnCcBnC,thenA~B
*(g) IfX c A, then X u (A n B) = (X u A) n B
(h) If A x B = A x C, then B = C
Throughout Exercises 3 through 6Jet U = {1,2,3,... ,9, 10). Part (a) of each
exercise calls for some experimentation with specific subsets of U, similar to that
required in Exercise 2.
- (a) Try to find a subsets A, B, and X of U with A and B distinct (i.e., A # B)
such that A u X = B u X and A n X = B n X. Do not try any more than five
combinations of the three sets.
(b) Suppose that three sets of the type described in (a) are impossible to find (i.e.,
do not exist). Can you formulate an elegant statement of a theorem asserting this
fact? (Note: If A = B, then surely A u X = B u X and A n X = B n X for any
set X). - (a) Try to find subsets A, B, and X of U with A and B distinct, such that
A n X = B n X and A n X' = B n X'. Do not try any more than five com-
binations of the three sets.
(b) Suppose, as in Exercise 3(b), that three sets satisfying the description in (a) do
not exist. Fomulate a well-stated theorem to this effect (noting that if A = B,
then A n X = B n X and A n X' = B n X' for any set X). - (a) Try to find subsets A, B, and Y of U, with A and B distinct and Y nonempty
such that A x Y = B x Y. Do not try any more than five combinations of the
three sets.
(b) Suppose again that three sets of the type described in (a) do not exist. For-
mulate a theorem that states this fact. - (a) Try to find subsets A and B of U such that either A c B and A n B' # 0
or A $ B and A n B' = 0. Do not try more than five combinations of the two
sets.
(b) Formulate an elegantly stated theorem describing the situation that two sets
of the type described in (a) do not exist. - (a) The equation a(b + c) = ab + ac, with a, b, and c real numbers, is the state-
ment that multiplication of real numbers distributes over addition:
(i) Write an equation that states that addition distributes over multiplication.
(ii) Is the equation you wrote for (i) true for all real numbers a, b, and c? (i.e.,
Does addition distribute over multiplication?) Give an example in support
of your answer.
*(b) Use the distributive lai(h) of Conjecture 3 to calculate the simultaneous solu-
tions to the inequalities in Example 3, Article 1.2. [Hint: Let X = (- m, -41
u [I, m), Y = (- m, -4), and Z = (5, a). The simultaneous solutions are the
elements of the set X n (Y u Z).] - Calculate the set of all real numbers that satisfy at least one of the following four
pairs of inequalities simultaneously:
(a) IX - $1 > and x2 - 9x - 22 > 0
(b) x2 - 3x - 28 1 0 and x2 - 9x - 22 1 0