,Contemporary Mathematics
Chapter 1
Sets
Solutions
1.1 Basic Set Concepts
YOUR TURN
1.1 1. Answers may vary. One possible solution: T = {wrench, screwdriver, hammer, plyers}
1.2 1. This is not well-defined since “medium-sized” is relative and subject to opinion.
2. This is well-defined.
1.3 1. ∅ or {}. No numbers are divisible by 0.
1.4 1. D = {0,1,2 ,…,9 }
1.5 1. M = {1,3,5 , … }
1.6 1. C = { c | c is a car }
1.7 1. I = { i | i is a musical instrument } Since listing out every musical instrument is a
tedious, and perhaps difficult task, the solution should have this form (set builder).
1.8 1. P = {} since 2 is the first prime number. Thus, n(P) = 0.
2. There are 26 lower-case letters in the English Alphabet. Thus, n(A) = 26.
1.9 1. We can count up the elements of B. Thus, B is finite.
2. We cannot count up the number of elements in the real numbers and ever finish. Thus, ℝ
is infinite.
1.10 1. B = { b , a , k , e } and A = { a , b , e , k } both have the same elements. Thus B = A.
2. B = { b , a , k , e } and F = { f , l , a , k , e } do not have the same number of elements. Thus,
they are neither equivalent nor equal.
3. B = { b , a , k , e } and C = { c , a , k , e } both have the same number of elements, but the
elements differ. (B has a “b” and C has a “c”.) Thus, B ∼ C .
CHECK YOUR UNDERSTANDING
1. set
2. cardinality
3. This is not a well-defined set. Whether a restaurant is in the top five is a matter of opinion, not a
fact.
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9/15/26
, Contemporary Mathematics
4. The cardinality of this set is 12.
5. n(A) = 12 and n(B) = 12. However, apples and donuts are different. Thus, A is equivalent to B but
they are not equal. We can write this as: A ~ B, but A ≠ B.
6. If all of the butterflies could be gathered in one place, we could count them up. This set is finite.
7. Roster method: { A , B ,C , … , Z }
Set builder notation: { x | x is an upper-case letter of the English alphabet}
EXERCISES
1. Let P represent the set. Then, P = {red, yellow, blue }
2. Let F represent the set. Then, F = {rose, tulip, marigold, iris, lily }.
3. Let A represent the set. Then, A = {50,51,52 , …,100 }.
4. Let B represent the set. Then, B = {18,19,20 ,… }
5. Let C represent the set. Then, C = {king, queen, rook, knight, bishop, pawn }
6. Let S represent the set. Then, S = {1,2,3 ,…,20 }.
7. Let L represent the set. Then, L = {l | l is a lizard}
8. Let S represent the set. Then, S = {s | s is a star}
9. Let M represent the set. Then, M = {3 n | n is a member of ℕ ¿
10. Let M represent the set. Then, M = {4 n | n is a natural number }.
11. Let P represent the set. Then, P = { p | p is an edible plant}
12. Let E represent the set. Then, E = {2 z | z is an integer}.
13. ∅ , since no squares are circles.
14. { }, since division by zero is undefined.
15. Let C represent the set. Then, C = {Greg, Peter, Bobby, Marsha, Jan, Cindy }
16. ℝ= { x | x is a real number }
17. ∅ , since no polar bears live in Antarctica.
18. Let S represent the set. Then, S = {s | s is a song written by Prince }. (You could write out all
the song titles and use the roster method, but this would be a very long list. Try to choose the
shortest method for writing a set that still makes the set well-defined.)
19. Let B represent the set.
B = {b | b is a children's book written and illustrated by Mo Willems } (You could write out
all the book titles and use the roster method, but this would be a very long list. Try to choose
the shortest method for writing a set that still makes the set well-defined.)
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, Contemporary Mathematics
20. Let C represent the set. Then, C = {red, orange, yellow, green, blue, indigo,violet }
21. Because we can list out all of the character names in the book, this is a well-defined set.
22. Because “greatest” is an opinion, and because “all time” depends upon when you are looking at
this list, this set is not well-defined.
23. The size of the group, the definition of “old”, and the definition of “new tricks” is not given and
are all subject to opinion. This set is not well-defined.
24. This is a well-defined set. A list of all the movies directed by Spike Lee as of 2021 could be made.
25. Since zebras cannot fly an airplane, this is the empty set. Thus, this is a well-defined set.
26. Since we can list out all of the names of the players in the group of National Baseball League Hall
of Fame members who have hit over 700 career home runs, this is a well-defined set.
27. n( P ) = 6
28. n(T ) = 8
29. n( ∅ ) = 0
30. n( B ) = 16
31. n( F ) = 9
32. n( { }) = 0
33. n(C ) = n(ℕ)=ℵ 0
34. n( S ) = n(ℕ)=ℵ 0
35. n( L ) = 14
36. The numbers on a standard six-sided die are 1, 2, 3, 4, 5, 6. Thus, the cardinality of this set is 6.
37. n( A) = n( { right, acute, obtuse } ) = 3; n( B ) = n( {equilateral, scalene, isoceles } ) = 3
Since both sets contain 3 elements, the two sets are equivalent. But the elements of the sets are
not the same, so the two sets are not equal.
Thus, A ~ B
38. n ( A ) = n
({
1,
1 1 1
, ,
2 3 4 })= 4 n( B ) = n({14 , 13 , 12 ,1})= 4.
; Since both sets contain the same
number of elements, they are equivalent. They both also contain the exact same elements, so they
are equal.
Thus, A = B.
39. n( A) = n ( { red, orange, yellow }) = 3; n( B ) = n( { green, blue, indigo, violet } ) = 4. Since they
do not contain the same number of elements, they are neither equivalent nor equal.
40. n( A) = n( {5 n | n ∈ ℕ } ) = ℵ 0; n( B ) = n(ℕ)=ℵ 0. The two sets do not contain the same
elements since A only contains the multiples of 5 and B contains every natural number.
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9/15/26
Chapter 1
Sets
Solutions
1.1 Basic Set Concepts
YOUR TURN
1.1 1. Answers may vary. One possible solution: T = {wrench, screwdriver, hammer, plyers}
1.2 1. This is not well-defined since “medium-sized” is relative and subject to opinion.
2. This is well-defined.
1.3 1. ∅ or {}. No numbers are divisible by 0.
1.4 1. D = {0,1,2 ,…,9 }
1.5 1. M = {1,3,5 , … }
1.6 1. C = { c | c is a car }
1.7 1. I = { i | i is a musical instrument } Since listing out every musical instrument is a
tedious, and perhaps difficult task, the solution should have this form (set builder).
1.8 1. P = {} since 2 is the first prime number. Thus, n(P) = 0.
2. There are 26 lower-case letters in the English Alphabet. Thus, n(A) = 26.
1.9 1. We can count up the elements of B. Thus, B is finite.
2. We cannot count up the number of elements in the real numbers and ever finish. Thus, ℝ
is infinite.
1.10 1. B = { b , a , k , e } and A = { a , b , e , k } both have the same elements. Thus B = A.
2. B = { b , a , k , e } and F = { f , l , a , k , e } do not have the same number of elements. Thus,
they are neither equivalent nor equal.
3. B = { b , a , k , e } and C = { c , a , k , e } both have the same number of elements, but the
elements differ. (B has a “b” and C has a “c”.) Thus, B ∼ C .
CHECK YOUR UNDERSTANDING
1. set
2. cardinality
3. This is not a well-defined set. Whether a restaurant is in the top five is a matter of opinion, not a
fact.
For more free, peer-reviewed, openly licensed resources visit OpenStax.org.
9/15/26
, Contemporary Mathematics
4. The cardinality of this set is 12.
5. n(A) = 12 and n(B) = 12. However, apples and donuts are different. Thus, A is equivalent to B but
they are not equal. We can write this as: A ~ B, but A ≠ B.
6. If all of the butterflies could be gathered in one place, we could count them up. This set is finite.
7. Roster method: { A , B ,C , … , Z }
Set builder notation: { x | x is an upper-case letter of the English alphabet}
EXERCISES
1. Let P represent the set. Then, P = {red, yellow, blue }
2. Let F represent the set. Then, F = {rose, tulip, marigold, iris, lily }.
3. Let A represent the set. Then, A = {50,51,52 , …,100 }.
4. Let B represent the set. Then, B = {18,19,20 ,… }
5. Let C represent the set. Then, C = {king, queen, rook, knight, bishop, pawn }
6. Let S represent the set. Then, S = {1,2,3 ,…,20 }.
7. Let L represent the set. Then, L = {l | l is a lizard}
8. Let S represent the set. Then, S = {s | s is a star}
9. Let M represent the set. Then, M = {3 n | n is a member of ℕ ¿
10. Let M represent the set. Then, M = {4 n | n is a natural number }.
11. Let P represent the set. Then, P = { p | p is an edible plant}
12. Let E represent the set. Then, E = {2 z | z is an integer}.
13. ∅ , since no squares are circles.
14. { }, since division by zero is undefined.
15. Let C represent the set. Then, C = {Greg, Peter, Bobby, Marsha, Jan, Cindy }
16. ℝ= { x | x is a real number }
17. ∅ , since no polar bears live in Antarctica.
18. Let S represent the set. Then, S = {s | s is a song written by Prince }. (You could write out all
the song titles and use the roster method, but this would be a very long list. Try to choose the
shortest method for writing a set that still makes the set well-defined.)
19. Let B represent the set.
B = {b | b is a children's book written and illustrated by Mo Willems } (You could write out
all the book titles and use the roster method, but this would be a very long list. Try to choose
the shortest method for writing a set that still makes the set well-defined.)
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9/15/26 1
, Contemporary Mathematics
20. Let C represent the set. Then, C = {red, orange, yellow, green, blue, indigo,violet }
21. Because we can list out all of the character names in the book, this is a well-defined set.
22. Because “greatest” is an opinion, and because “all time” depends upon when you are looking at
this list, this set is not well-defined.
23. The size of the group, the definition of “old”, and the definition of “new tricks” is not given and
are all subject to opinion. This set is not well-defined.
24. This is a well-defined set. A list of all the movies directed by Spike Lee as of 2021 could be made.
25. Since zebras cannot fly an airplane, this is the empty set. Thus, this is a well-defined set.
26. Since we can list out all of the names of the players in the group of National Baseball League Hall
of Fame members who have hit over 700 career home runs, this is a well-defined set.
27. n( P ) = 6
28. n(T ) = 8
29. n( ∅ ) = 0
30. n( B ) = 16
31. n( F ) = 9
32. n( { }) = 0
33. n(C ) = n(ℕ)=ℵ 0
34. n( S ) = n(ℕ)=ℵ 0
35. n( L ) = 14
36. The numbers on a standard six-sided die are 1, 2, 3, 4, 5, 6. Thus, the cardinality of this set is 6.
37. n( A) = n( { right, acute, obtuse } ) = 3; n( B ) = n( {equilateral, scalene, isoceles } ) = 3
Since both sets contain 3 elements, the two sets are equivalent. But the elements of the sets are
not the same, so the two sets are not equal.
Thus, A ~ B
38. n ( A ) = n
({
1,
1 1 1
, ,
2 3 4 })= 4 n( B ) = n({14 , 13 , 12 ,1})= 4.
; Since both sets contain the same
number of elements, they are equivalent. They both also contain the exact same elements, so they
are equal.
Thus, A = B.
39. n( A) = n ( { red, orange, yellow }) = 3; n( B ) = n( { green, blue, indigo, violet } ) = 4. Since they
do not contain the same number of elements, they are neither equivalent nor equal.
40. n( A) = n( {5 n | n ∈ ℕ } ) = ℵ 0; n( B ) = n(ℕ)=ℵ 0. The two sets do not contain the same
elements since A only contains the multiples of 5 and B contains every natural number.
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9/15/26
