240
Sets, Combinatorics, and Probability
5. Describe each of the following sets by listing its elements:
a. {x 0 x [ ℕ and x
2
< 25}
b. {x 0 x [ ℕ and x is even and 2 < x < 11}
c. {x 0 x is one of the first three U.S. presidents}
d. {x 0 x [ ℝ and x
2
= −1}
e. {x 0 x is one of the New England states}
f. {x 0 x [ ℤ and 0 x 0 < 4} ( 0 x 0 denotes the absolute value function)
6. Describe each of the following sets by listing its elements:
a. {x 0 x [ ℕ and x
2
− 5x + 6 = 0}
b. {x 0 x [ ℝ and x
2
= 7}
c. {x 0 x [ ℕ and x
2
− 2x − 8 = 0}
7. Describe each of the following sets by giving a characterizing property:
a. {1, 2, 3, 4, 5}
b. {1, 3, 5, 7, 9, 11, …}
c. {Melchior, Gaspar, Balthazar}
d. {0, 1, 10, 11, 100, 101, 110, 111, 1000, …}
8. Describe each of the following sets:
a {x 0 x [ ℕ and (E q)(q [ {2, 3} and x = 2q)}
b. {x 0 x [ ℕ and (E y)(E z)(y [ {0, 1} and z [ {3, 4} and y < x < z)}
c. {x 0 x [ ℕ and (4y)(y even S x ∙ y)}
9. Given the description of a set A as A = {2, 4, 8 …}, do you think 16 [ A?
10. What is the cardinality of each of the following sets?
a. S = {a, {a, {a}}}
b. S = {{a}, {{a}}}
c. S = {[}
d. S = {a, {[}, [}
e. S = {[, {[, {[}}, {[, {[, {[}}}}
11. Let
A = {2, 5, 7}
B = {1, 2, 4, 7, 8}
C = {7, 8}
Which of the following statements are true?
a. 5 # A
d. 7 [ B
b. C # B
e. {2, 5} # A
c. [ [ A
f. [ # C
12. Let
A = {x 0 x [ ℕ and 1 < x < 50}
B = {x 0 x [ ℝ and 1 < x < 50}
C = {x 0 x [ ℤ and 0 x 0 ≥ 25}