242
Sets, Combinatorics, and Probability
17. Let
A = {(x, y) 0 (x, y) lies within 3 units of the point (1, 4)}
and
B = {(x, y) 0 (x − 1)
2
+ (y − 4)
2
≤ 25}
Prove that A ( B.
18. Let
A = {x 0 x [ ℝ and x
2
− 4x + 3 < 0}
and
B = {x 0 x [ ℝ and 0 < x < 6}
Prove that A ( B.
19. Program QUAD finds and prints solutions to quadratic equations of the form ax
2
+ bx + c = 0. Program
EVEN lists all the even integers from −2n to 2n. Let Q denote the set of values output by QUAD and E
denote the set of values output by EVEN.
a. Show that for a = 1, b = −2, c = −24, and n = 50, Q # E.
b. Show that for the same values of a, b, and c, but a value for n of 2, Q h E.
20. Let A = {x 0 cos(x/2) = 0} and B ={x 0 sin x = 0}. Prove that A # B.
21. Which of the following statements are true for all sets A, B, and C?
a. If A # B and B # A, then A = B.
d. [ [ {[}
b. {[} = [
e. [ # A
c. {[} = {0}
22. Which of the following statements are true for all sets A, B, and C?
a. [ [ A
b. {[} = {{[}}
c. If A ( B and B # C, then A ( C.
d. If A ∙ B and B ∙ C, then A ∙ C.
e. If A [ B and B h C, then A o C.
23. Prove that if A # B and B # C, then A # C.
24. Prove that if A′ # B′, then B # A.
25. Prove that for any integer n ≥ 2, a set with n elements has n (n − 1)/2 subsets that contain exactly two
elements.
26. Prove that for any integer n ≥ 3, a set with n elements has n(n − 1)(n − 2)/6 subsets that contain exactly
three elements. (Hint: Use Exercise 25.)
27. Find ℘(S) for S = {a}.
28. Find ℘(S) for S = {a, b}.
29. Find ℘(S) for S = {1, 2, 3, 4}. How many elements do you expect this set to have?
30. Find ℘(S) for S = {[}.