HW1 (Set Theory) Date:__________, Name___________________________________
You need Scantron 882E. Please use a pencil to mark the answers. Make sure your Scantron is clean , flat, and not folded
when you submit. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
List the elements in the set.
1) {x | x is a whole number between 1 and 5}
A) {1, 2, 3, 4, 5}
B) {2, 3, 4}
1)
C) {1, 2, 3, 4}
D) {2, 3, 4, 5}
C) {3, 4, 5, 6, 7}
D) {4, 5, 6, 7}
2)
2) {x | x is an integer between 3 and 7}
B) {4, 5, 6}
A) {3, 4, 5, 6}
3) {x | x is a negative multiple of 5}
A) {0, -5, -10, ...}
C) {5, 10, 15, ...}
3)
B) {-5, -10, -15, ...}
D) {-5, -25, -125, ...}
4)
4) The set of all whole numbers greater than 3 and less than 7
B) {3, 4, 5, 6, 7}
C) {4, 5, 6}
A) {3, 4, 5, 6}
D) {4, 5, 6, 7}
5) {x | x is a counting number less than -4}
B) ∅
A) {-3, -2, -1, ...}
D) {-5, -6, -7, ...}
5)
C) {..., -7, -6, -5}
6) The set of all positive integer powers of 3.
A) {1, 3, 9, 27, 81, 243, ...}
C) {1, 8, 27, 64, 125, ...}
B) {3, 6, 9, 12, 15, ...}
D) {3, 9, 27, 81, 243, ...}
7) { x x is an even integer smaller than 8}
A) {..., -6, -4, -2, 0, 2, 4, 6}
C) {..., -6, -4, -2, 2, 4, 6}
B) {2, 4, 6}
D) {0, 2, 4, 6}
6)
7)
8) The set of the days of the week
A) {Sunday, Monday, Tuesday, Wednesday, Thursday,
Friday, Sunday}
B) {Tuesday, Thursday}
C) {Saturday, Sunday}
D) {Friday, Monday, Saturday, Sunday, Thursday,
Tuesday, Wednesday}
Write the set in set-builder notation.
9) {2}
A) {x is a constant}
C) {x | x is a natural number}
8)
9)
B) {x | x is the natural number 2}
D) {x}
10) {2, 4, 6, 8}
A) {x | x is any even integer less than 10}
B) {x | x is an even natural number less than 10}
C) {2, 4, 6, 8}
D) {x | x is any even natural number}
10)
11) {17, 18, 19, 20}
A) {x | x is an integer between 17 and 20}
C) {x | x is an integer between 16 and 21}
11)
B) {x | x is an integer less than 21}
D) {17, 18, 19, 20}
1
12) {-6, -5, -4, -3, ...}
A) {x | x is an integer greater than -7}
C) {x | x is an integer between -7 and -2}
B) {x | x is any integer}
D) {-6, -5, -4, -3}
13) {..., -3, -2, -1, 0, 1, 2, 3, ...}
A) {-3, -2, -1, 0, 1, 2, 3}
C) {x | x is a natural number}
B) {x | x is any integer greater than -3}
D) {x | x is an integer}
12)
13)
14)
14) {2, 4, 8, 16, 32, ...}
A) {x x is an integer power of 2}
B) {x x is a positive multiple of 2}
C) {x x is a positive integer power of 2}
D) {x x is a positive multiple of 4}
Identify the set as finite or infinite.
15) {9, 10, 11, ..., 36}
A) Finite
15)
B) Infinite
1 1 1
16) 1, , , , ...
3 9 27
16)
A) Finite
B) Infinite
17) {x | x is a counting number larger than 480}
A) Finite
17)
B) Infinite
18) {x | x is an odd counting number}
A) Finite
B) Infinite
18)
5 25 125
3125
19) 1, , , , ..., 7 49 343
16807
19)
A) Infinite
Find n(A) for the set.
20) A = {2, 4, 6, 8, 10}
A) n(A) = 5
B) Finite
20)
B) n(A) = 2
C) n(A) = 10
D) n(A) = 4
21) A = {300, 301, 302, ..., 3000}
A) n(A) = 2700
B) n(A) = 2701
C) n(A) = 3000
D) n(A) = 4
22) A = {x | x is a month in the year}
A) n(A) = 12
B) n(A) = 1
C) n(A) = 24
D) n(A) = 52
21)
22)
23) A = {x | x is a number on a clock face}
A) n(A) = 6
B) n(A) = 24
23)
C) n(A) = 12
D) n(A) = 3
24) A = {x | x is a second in a minute}
A) n(A) = 120
B) n(A) = Infinite
C) n(A) = 60
D) n(A) = 12
24)
2
Determine whether or not the set is well defined.
25) {x | x is a tennis player who has won at Wimbledon}
A) Not well defined
B) Well defined
26) {x | x is a low-fat ice cream}
A) Well defined
25)
26)
B) Not well defined
27) {x | x is a football team that has won the Super Bowl}
A) Well defined
B) Not well defined
27)
28) {x | x is a adventure book in the library}
A) Not well defined
28)
B) Well defined
29) {x | x is a stock on the AmEx today}
A) Not well defined
B) Well defined
29)
Complete the blank with either ∈ or ∉ to make the statement true.
30) -8 {8, 10, 12, ..., 20}
A) ∈
B) ∉
31) 0 {-1, 1, 3, 15, 25}
A) ∈
B) ∉
32) {8} {{5}, {6}, {7}, {8}, {9}}
A) ∈
B) ∉
33) 5 {11, 10, 9, 8}
A) ∉
B) ∈
34) 7 {3, 12, 6, 7, 12}
A) ∈
B) ∉
35) 9 {7, 8, 9, 10}
A) ∈
B) ∉
30)
31)
32)
33)
34)
35)
Tell whether the statement is true or false.
36) 6 ∈ {12, 18, 24, 30, 36}
A) True
36)
B) False
37) {2, 6, 12} = {0, 2, 6, 12}
A) True
37)
B) False
38) 17 ∉ {16, 14, 13, ..., 1}
A) True
38)
B) False
39) {6} = {x | x is an even counting number between 8 and 14}
A) True
B) False
3
39)
40) {57, 58, 57, 58} = {57, 58}
A) True
40)
B) False
Write true or false for the following statement.
Let A = {3, 5, 7, 9, 11, 13} B = {3, 5, 9, 11} C = {5, 9, 13}
41) 9 ∉ C
A) True
B) False
41)
42) 9 ∈ B
A) True
B) False
43) Every element of B is also an element of C.
A) True
B) False
42)
43)
44) A = {x | x is an odd counting number greater than 1 and less than 15}
A) True
B) False
44)
45) 0 ∈ A
A) True
45)
B) False
46) Every element of C is also an element of A.
A) True
B) False
46)
47) {x | x is an odd counting number less than 15} = A
A) True
47)
B) False
Use ⊆ or ⊈ in the blank to make a true statement.
48) {6, 8, 10} {5, 6, 7, 8, 10}
A) ⊆
48)
B) ⊈
49) {5, 23, 28} {6, 23, 28, 38}
A) ⊆
B) ⊈
49)
50) {c, d, j, f} {c, d, j, f, n}
A) ⊈
B) ⊆
51) ∅ ∅
A) ⊈
B) ⊆
52) {11, 13, 15} {x | x is an odd counting number}
A) ⊈
B) ⊆
53) {b, p, a} {b, b, p, p, a, a}
A) ⊈
B) ⊆
50)
51)
52)
53)
Decide whether ⊆, ⊂, both, or neither can be placed in the blank to make a true statement.
54) {11, 12, 13} {10, 11, 12, 13}
A) ⊆
B) Both ⊂ and ⊆
C) Neither
D) ⊂
4
54)
55) ∅ {3, 17, 30, 42}
A) Both ⊂ and ⊆
B) ⊂
C) Neither
D) ⊆
56) {15, 16, 17} {15, 16, 17}
A) Neither
B) ⊆
C) Both ⊂ and ⊆
D) ⊂
55)
56)
57)
57) {0} ∅
A) Neither
B) ⊂
C) Both ⊂ and ⊆
D) ⊆
58) {a, b} {z, a, y, b, x, c}
A) ⊆
B) Both ⊂ and ⊆
C) ⊂
D) Neither
59) {s, r, t} {s, r, t}
A) ⊆
B) Both ⊆ and ⊂
C) ⊂
D) Neither
58)
59)
Determine whether the statement is true or false.
Let A = {1, 3, 5, 7} B = {5, 6, 7, 8} C = {5, 8} D = {2, 5, 8} U = {1, 2, 3, 4, 5, 6, 7, 8}
60) C ⊂ D
A) True
B) False
60)
61) ∅ ⊆ A
A) True
B) False
61)
62) {6, 5, 8, 7} ⊆ B
A) True
62)
B) False
63) D ⊆ B
A) True
B) False
64) A ≠ {7, 5, 3, 1}
A) True
B) False
63)
64)
65) {5} ⊆ D
A) True
65)
B) False
66) {0} ⊆ U
A) True
B) False
67) {8, 5, 2} ⊂ D
A) True
B) False
66)
67)
Decide whether ⊆, ⊂, both, or neither can be placed in the blank to make a true statement.
68) {5, 6, 7} {4, 5, 6, 7}
A) ⊆
B) Neither
C) Both ⊂ and ⊆
D) ⊂
69) ∅ {2, 12, 25, 33}
A) ⊂
68)
69)
C) ⊆
B) Both ⊂ and ⊆
5
D) Neither
70) {13, 14, 15} {13, 14, 15}
A) Both ⊂ and ⊆
70)
B) ⊆
C) ⊂
D) Neither
Find the number of subsets of the set.
71) {6, 7, 8}
A) 7
B) 8
C) 3
D) 6
72) {x | x is an even number between 13 and 27}
A) 6
B) 64
C) 40
D) 128
73) {0}
A) 2
C) 1
D) 0
C) 12
D) 14
71)
72)
73)
B) 4
74) {mom, dad, son, daughter}
A) 16
B) 8
74)
75) {math, English, history, science, art}
A) 32
B) 16
75)
C) 24
D) 28
C) 7
D) 2
77) {x | x is an even number between 15 and 29}
A) 28
B) 127
C) 64
D) 128
78) {0}
A) 0
B) 2
C) 1
D) 4
79) {car, boat, truck, train}
A) 8
B) 16
C) 15
D) 14
Find the number of proper subsets of the set.
76) {3, 4, 5}
A) 5
B) 6
76)
77)
78)
79)
Let U = {1, 2, 4, 5, a, b, c, d, e}. Find the complement of the set.
80) Q = {2, 4, b, d}
A) {1, 2, 4, 5, a, b, c, d, e}
B) {1, 5, a, e}
C) {1, 3, 5, a, c, e}
D) {1, 5, a, c, e}
80)
81) W = {1, 5, e, d, a}
A) {2, 3, 4, a, b, c}
B) {2, 4, b, c}
C) {2, 3, 4, b, c}
D) {1, 2, 4, b, c}
81)
82) V = {1, 2, 4, 5, a, b, c, e}
A) {u}
B) {3, d}
C) {d}
D) ∅
83) T = {a, b, c, d}
A) {1, 2, 4, 5}
B) {1, 2, 3, 4, 5, e}
C) {e}
D) {1, 2, 4, 5, e}
82)
83)
84) M = {a}
A) {1, 2, 3, 4, 5, b, c, d, e}
C) {1, 2, 4, 5, b, c, d, e}
84)
B) {u, v}
D) {1, 2, 5, b, c, d, e}
6
85)
85) S = ∅
A) ∅ʹ
B) U
C) {0}
D) ∅
86) P = {a, b, d, e, 1, 2, 4, 5}
A) U
B) {c, 3}
C) ∅
D) {c}
86)
The lists below show five agricultural crops in Alabama, Arkansas, and Louisiana.
Alabama
soybeans (s)
peanuts (p)
corn (c)
hay (h)
wheat (w)
Arkansas
soybeans (s)
rice (r)
cotton (t)
hay (h)
wheat (w)
Louisiana
soybeans (s)
sugarcane (n)
rice (r)
corn (c)
cotton (t)
Let U be the smallest possible universal set that includes all of the crops listed, and let A, K and L be the sets of five
crops in Alabama, Arkansas, and Louisiana, respectively. Find each of the following sets.
87)
87) The set of crops in U.
A) {s, p, c, h, w, s, r, t, h, w, s, n, r, c, t}
B) {s, p, c, h, w, r, t, n, c}
C) {c, h, n, p, r, s, t, w}
D) {s, p, c, w, r, t, n}
88) The set of crops in Aʹ.
A) {c, h, n, r, s, t, w}
C) {h, n, r, t}
88)
B) {n, r, t}
D) {r, t}
89) The set of crops in both A and K
A) {c, p, r, t}
C) {c, h, p, r, s, t, w}
B) {c, h, s, t, w}
D) {h, s, w}
90) The set of crops in both L and K
A) {c, n, r, s, t}
C) {c, h, n, w}
B) {c, h, n, r, s, t, w}
D) {r, s, t}
89)
90)
91) The set of crops in both L and Kʹ
A) {c, n}
B) {r, s, t}
91)
C) {h, w}
D) {c, n, p}
92) The set of crops in both A and Lʹ
A) {n, r, t}
B) {c, s}
C) {h, p, w}
D) {h, n, t, w}
93) The set of crops in both Aʹ and Kʹ
A) {c, n, p, r, t}
B) {n}
C) ∅
D) {c, p, r, t}
92)
93)
94) The set of crops common to A, K, and L
A) {n, p, s}
C) {n, p}
B) {s}
D) {c, h, n, p, r, s, t, w}
95) The set of crops in either A or L or both
A) {c, h, n, p, r, s, t, w}
C) {h, n, p, r, t, w}
B) {c, n, p}
D) {c,s}
94)
95)
7
96) The set of crops in either Aʹ or L or both
A) {h, p, w}
B) {c, n, r, s, t}
96)
C) {h, n, p, r, t, w}
Solve the problem.
97) List all possible subsets of the set {m, n}.
A) {m}, {n}
C) {m}, {n}, {m, n}, ∅
D) {n, r, t}
97)
B) {m}, {n}, ∅
D) {m}, {n}, {m, n}
98) List all possible proper subsets of the set {2, 6, 7}.
A) {2}, {6}, {7}, {2, 6}, {2, 7}, {6, 7}, {2, 6, 7}
C) ∅, {2}, {6}, {7}, {2, 6}, {2, 7}, {6, 7}
98)
B) ∅, {2}, {6}, {7}, {2, 6}, {2, 7}, {6, 7}, {2, 6, 7}
D) {2}, {6}, {7}, {2, 6}, {2, 7}, {6, 7}
99) A committee is to be formed. Possible candidates for the committee are Eric, Frances, Greg, and
Jose. Denoting these four people by e, f, g, j, list all possible committees of two people (ie list all
possible subsets of size two).
A) {e, f}, {e, g}, {e, j}, {f, j}, {g, j}
B) {e, f}, {e, g}, {e, j}, {f, g}, {f, j}, {g, j}, {f, e}, {g, e}
C) {e, f}, {e, g}, {f, g}, {g, j}
D) {e, f}, {e, g}, {e, j}, {f, g}, {f, j}, {g, j}
99)
100) A committee is to be formed. Possible candidates for the committee are Eric, Frances, Greg, and
Jose. Denoting these four people by e, f, g, j, list all possible committees if the committee is to
contain at least two people and may contain up to four people.
A) {e, f}, {e, g}, {e, j}, {f, g}, {f, j}, {g, j}, {e, f, g}, {e, f, j}, {e, g, j}, {f, g, j}, {e, f, g, j}
B) {e, f}, {e, g}, {e, j}, {f, j}, {e, f, g}, {e, f, j}, {e, g, j}, {f, g, j}, {e, f, g, j}
C) {e, f}, {e, g}, {e, j}, {f, g}, {f, j}, {g, j}, {e, f, g}, {e, f, j}, {f, g, j}, {e, f, g, j}
D) {e, f}, {e, g}, {e, j}, {f, g}, {f, j}, {g, j}, {e, f, g}, {e, f, j}, {e, g, j}, {f, g, j}
100)
8