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Introduction to Sets
Some Exercises Solutions
1. List the members of the following sets. Sketch their graphs on a Real Number line
a) { x | x is a real number such that x² = 4}= {-2, 2}
b) { x | x is an integer such that x² = 2} = 
c) { x | x is a positive integer such that x < 2} = {1}
d) { x | x is a positive integer such that x < 2 or 4 < x < 7} = {1, 5, 6}
e) { x | x is a positive integer such that x < 2 and 4 < x < 7} = 
2. Determine whether each of the following pairs of sets is equal.
a) {1, 2, 1, 3, 1, 2}, {2, 3, 1} They are equal {1, 2, 1, 3, 1, 2} = {2, 3, 1}
b) {{1}}, {1, {1}} They are not equal {{1}}  {1, {1}}
c)
, { } They are not equal   { }
3. Write each of the following sets with set builder notation. Sketch their graphs on a Real Number line.
a) {1,2,3,4} = {x | x  N and x < 5} = {x | x  N and x ≤ 4}= {x | x  Z and 0 < x < 5}
={x | x  Z and 1 ≤ x ≤ 4}
b) {2, 5, 8, 11, 14} = {x | x  N and x =3k + 2 for k = 0, 1, 2, 3, 4}
c) {-3, -2, -1, 0, 1, 2, 3} = {x | x  Z and |x| < 3} = {x | x  Z and -3 < x < 3}
d) (-4, 8) = {x | x  R and -4 < x < 8}
e) (3, 10] = {x | x  R and 3 < x ≤ 10}
f) [-14, 2) = {x | x  R and -14 ≤ x < 2}
g) [0, 3] = {x | x  R and 0 ≤ x ≤ 3} (-2, 0)(0,1) = {x  R | -2 < x < 0 or 0 < x < 1}
h) (-2, 0) (0,1) = 
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i) [-2, 0](0,1] = {x  R | -2 ≤ x ≤ 1}
j) (-∞, 5)(7,+∞) = {x  R | x < 5 or x > 7}
4. Write each of the following sets with the roster method. Sketch their graphs on a Real Number line.
a) {x| x  N and x < 8} = {1, 2, 3, 4, 5, 6, 7}
b) {x| x  N and -3 < x < -1} = 
c) {x| x  Z and -3 < x < -1} = {- 2}
d) {x| x  W and -3 < x < -1} = 
e) {x| x  Z and -3 < x < -1 or x  N} = {-2, 1, 2, 3, …} There is no really good way to “graph”
this set
5. Write each of the following sets with interval notation. Sketch their graphs on a Real Number line.
How many elements are in each set?
a) {x| x  R and x < 8} = (-∞, 8) There are infinitely many elements in the set.
b) {x| x  R and (x < 8 or x > 12)}={ x  R | x < 8 or x > 12} = (-∞, 8)(12, ∞)
There are infinitely many elements in the set.
c) {x| x  R and 3 < x < 8} = {x  R|3 < x and x < 8}= { x  R|3 < x < 8} = (3, 8)
There are infinitely many elements in the set.
d) {x  R | x < -3 or x > 1}= (-∞, -3)(1, ∞) There are infinitely many elements in the set.
e) {x  R | x < 5 or x > 0}= (-∞, 5)(0, ∞) There are infinitely many elements in the set.
f) {x  R | x > 5 or x > 0}= (0, ∞) There are infinitely many elements in the set.
g) { x  R | x < 5 or x < 0}= (-∞, 5) There are infinitely many elements in the set.
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h) {x  R|3 < x and x > 0}= (3, ∞) There are infinitely many elements in the set.
i) {x  R|1 < x and x < 3}= (1, 3) There are infinitely many elements in the set.
j) {x  R|-2 < x and x < 1}= (-2, 1) There are infinitely many elements in the set.
k) {x  R|3 > x and x < 8}= (-∞, 3) There are infinitely many elements in the set.
l) {x  Q|-2 < x and x < 1} F =  There are no elements in the set.
Let A = {1, 2, 3, 4, 5, 6, 7}
Let B = { 1, 3, 5, 6}
Let C = {2, 4, 6, 8, 10}
Let D = {1, 2, 6, 7}
m) Write all of the subset relations that exist between A, B, C, and D. B  A and D  A
n) Compute A  B. Draw a Venn diagram to illustrate. A  B =
o) Compute A  B Draw a Venn diagram to illustrate. A  B = B
p) Compute B  C  D Draw a Venn diagram to illustrate. B  C  D = {1,2,3,4,5,6,7,8,10}
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A
q) Compute B  C D Draw a Venn diagram to illustrate. B  C D = {6}
See Diagram for p)
r) Compute B  C Draw a Venn diagram to illustrate. B  C = {6}
See Diagram for p)
s) Compute D  (B  C) = {1, 2, 6} and compute (D  B)  (D  C) = {1, 6, 2}
t) Compute C   = C
u) Compute D   = 
6. Recall the meanings of the special symbols N, W, Z, Q, F, and R. Calculate each of the following sets.
Write the sets with the roster method, set builder notation, and interval notation whenever possible.
How many elements are in each set? Sketch the graph of each set on a Real number line. Use Venn
diagrams to show the relationships between the sets involved in each question.
a) N  (2,5) = {3, 4} = {x  N | 2 < x < 5} two elements Interval notation is no t possible
b) N  (-2,0] =  no elements Interval notation is not possible
c) N  [3,5] = {3, 4, 5} = {x  N | 3 ≤ x ≤ 5} three elements Interval notation is not possible
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d) Z  (2,5) = {3, 4} = {x  N | 2 < x < 5} = {x  Z | 2 < x < 5} two elements Interval notation is
not possible
e) Z  (-2,0] = {-1, 0} = {x  Z |-1 ≤ x ≤ 0} two elements Interval
notation is not possible
f) Z  [3,5] = {3, 4, 5} = {x  N | 3 ≤ x ≤ 5} = {x  Z | 3 ≤ x ≤ 5} three elements Interval
notation is not possible
g) R  (2,5) = (2, 5) = {x  R | 2 < x < 5} infinitely many
elements
h) R  (-2,0] = (-2, 0] = {x  R | 2 < x ≤ 0} infinitely many
elements
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i) R  [3,5] = [3, 5] = {x  R | 3 ≤ x ≤ 5} infinitely many elements
j) Q  (2,5) = {x  Q | 2 < x < 5} infinitely many elements Interval notation is not possible
Because It is impossible to show the rational numbers without
showing the irrational numbers on the number line, it is impossible
to illustrate this intersection of sets on the number line.
k) Q  (-2,0] = {x  Q | -2 < x < 0} infinitely many elements
Interval notation is not possible
Because It is impossible to show the rational numbers without
showing the irrational numbers on the number line, it is impossible
to illustrate this intersection of sets on the number line.
l) Q  [3,5] = {x  Q | 3 ≤ x ≤ 5} infinitely many elements Interval notation is not possible
Because It is impossible to show the rational numbers without showing
the irrational numbers on the number line, it is impossible to illustrate
this intersection of sets on the number line.
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For Problems 7 – 11 use the sets A, B, and C, as defined here.
A = {2, 3, 5, 7} Let B = {2, 4, 5, 6} Let C = { 12, 28, 35}.
7. The set {2, 5} is a subset of A. Using the subset symbol we can write this same fact much more
efficiently as 2,5  A . Use the subset symbol to make a similar statement for each subset of A.
  A, A  A,
{2}  A, {3}  A, {5}  A, {7}  A,
{2,3}  A, {2,5}  A, {2,7}  A, {3,5}  A, {3,7}  A, {5,7}  A,
{2,3,5}  A, {2,3,7}  A, {2,5,7}  A, {3,5,7}  A,
8. 4 is an element of B. Using the symbol for “is an element of” we can write this more efficiently by
writing 4  B .Use the symbol for “is an element of” to make a similar statement about each element of
B.
2  B, 4  B, 5  B, 6  B
9. What is A B ? How is it related to B? How is it related to A?
A  B = {2, 5}, A  B  A and A  B  B
10. What is B C ? How is it related to B? How is it related to C?
B  C = , B  C  B, and B  C  C
11. What is A B ? How is it related to A? How is it related to B?
A  B = {2, 3,4, 5,6, 7}, A  A  B, B  A  B
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LCD, LCM, GCD, and SETS
To insure that everyone understands all the words in the following problems, definitions of the necessary terms
are presented before the exercises. Many, maybe all, of these are familiar concepts.
Definition: The set of Natural Numbers consists of the common counting numbers.
Standard Symbol: N
Symbolic Form: N = {1, 2, 3, 4, … }
Definition: A prime number is a natural number greater than 1 which has only 1 and itself as factors.
Not Standard but Common Symbol: P
Definition: A composite number is a natural number greater than 1 which has more factors than 1 and itself.
Fact: The natural number 1 is neither a prime number nor a composite number.
Definition: A Natural Number n is divisible by a Natural Number m if there is a Natural Number k such that
n = km.
Definition: If a Natural Number n is divisible by a Natural Number m, then n is a multiple of m.
Definition: If a Natural Number n is divisible by a Natural Number m, then m is a divisor of n.
Definition: If a Natural Number n is divisor of two Natural Numbers a and b, then n is a common divisor of a
and b.
Definition:If a Natural Number n is the largest divisor of two Natural Numbers a and b, then n is the Greatest
Common Divisor (GCD) of a and b. Frequently symbolized by (a,b). Note the context must be clear before
using this notation!
Definition: If a Natural Number m is a multiple of two Natural Numbers n and q, then m is a common multiple
of n and q.
Definition: The smallest common multiple of two Natural Numbers n and q is the Least Common Multiple
(LCM) of n and q. Frequently symbolized by [a,b]. Note the context must be clear before using this notation!
Definition: The Least Common Denominator (LCD) of two fractions is the Least Common Multiple (LCM)
of their denominators.
Definition: A set is a collection of objects. Objects in the collection are called elements of the set.
Definition: The roster method of specifying a set consists of surrounding the collection of elements with
braces.
Extension: A variation of the simple roster method uses the ellipsis ( … ) when the pattern is obvious and the
set is large. For these exercises, I will call this the extended roster method.
Example: {1, 3, 5, 7, … , 9007} is the set of odd counting numbers less than or equal to 9007.
Exercise 1: Use the extended roster method to write the set of multiples of 3. Call this set M3.
M3 = {3, 6, 9, 12, 15, …}
Exercise 2: Use the extended roster method to write the set of multiples of 5. Call this set M5.
M5 = {5, 10, 15, 20, …}
Exercise 3: Use the extended roster method to write M3M5. Call this set M[3,5].
M[3,5] = {15, 30, 45, 60, …}
Exercise 4: How would you describe the elements of M[3,5]? Give two descriptions.
The elements of M[3,5] are multiples of both 3 and 5. They are common multiples of 3 and 5.
The elements of M[3,5] are multiples of 15, the smallest multiple of both 3 and 5.
Exercise 5: How would you name the smallest element of M[3,5]?
The smallest element of M[3,5] should be named the Least Common Multiple (LCM) of 3 and 5.
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Note that in Exercises 1 – 5, the numbers 3 and 5 are prime numbers. Primes are pretty special. You should
start wondering what these sets look like if the numbers are composites. What if one is prime and the other is
composite? What if the prime is a factor of the composite? How about two composites that have no common
factors? What about the case when they are both composite and do have common factors.
For example, using the above notation, you should look at the following sets (and others like these)
a) M3, M6, M[3,6] ,and its least element (LCM)
b) M3, M8, M[3,8] ,and its least element (LCM)
c) M6, M18, M[6,18] ,and its least element (LCM)
d) M10, M14, M[10,14] ,and its least element (LCM)
and so on – that is called studying.
Observe from the above that the LCM of 3 and 5 is 15 – the product of the two numbers 3 and 5.
Observe that 3 and 5 have no common factors.
M3 = {3, 6, 9, 12, 15, …} M6 = {6, 12, 18, 24, …} M[3,6] = {6, 12, 18, 24, …} = M6
LCM of 3 and 6 is 6 – the larger of the two numbers 3 and 6.
Observe that 6 is a multiple of 3.
M3 = {3, 6, 9, 12, 15, …} M8 = {8, 16, 24, 32, …} M[3,8] = {24, 48, 72, 96, …} = M24
LCM of 3 and 8 is 24 – the product of the two numbers 3 and 8.
Observe that 3 and 8 have no common factors.
M6 = {6, 12, 18, 24, …} M18 = {18, 36, 54, 72, …} M[3,6] = {18, 36, 54, 72, …} = M18
LCM of 6 and 18 is 18 – the larger of the two numbers 6 and 18.
Observe that 18 is a multiple of 6.
The four red flagged observations illustrate two general principles.
1) If two natural numbers have no common factors, then their LCM is their product.
2) If one natural number is a multiple of another natural number, then their LCM is the larger of the two.
M10 = {10, 20, 30, 40, 50, …} M14 = {14, 28, 42, 56, …} M[10,14] = {70, 140, 210, 280, …} = M70
LCM of 10 and 14 is 70 – not their product nor the largest of the two.
Observe that 10 and 14 have common factors but neither is a multiple of the other.
This last observation points to a need for further investigation of ways of computing the LCM of two numbers.
Such investigation is not within the scope of this course. Suffice it to say that the method of computing the
LCM of two numbers involves an examination of the prime factorization of each of the two numbers.
Exercise 6: Use the roster method to write the set of divisors of 24. Call this set D24.
D24 = {1, 2, 3, 4, 6, 8, 12, 24}
Exercise 7: Use the roster method to write the set of divisors of 15. Call this set D15.
D15 = {1, 3, 5, 15}
Exercise 8: Use the roster method to write D24D15. Call this set D(24,15).
D(24,15) = D24D15 = {1, 3}
Exercise 9: How would you describe the elements of D(24,15)?
They are divisors of both 15 and 24. They are common divisors of 15 and 24.
Exercise 10: How would you name the largest element of D(24,15)?
The largest element of D(24,15) should be named the Greatest Common Divisor (GCD) of 15 and 24.
Just as you did after Exercises 1-5, you should now look at similar combinations of numbers h and k and the
sets Dk, Dh, and D(h,k).
For example, using the above notation, you should look at the following sets (and others like these)
a) D12, D 35, D (12,35) and its greatest element
b) D9, D 8, D (9,8) and its greatest element
c) D19, D31, D (19,31) and its greatest element
d) D10, D75, D (10,75) and its greatest element
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and so on – that is called studying.
D12 = {1, 2, 3, 4, 6, 12} D 35 = {1, 5, 7, 35} D (12,35) = {1} The greatest element is 1.
D9 = {1, 3, 9} D 8 = {1, 2, 4, 8} D (9,8) = {1} The greatest element is 1.
D19 = {1, 19} D 35 = {1, 5, 7, 35}
D (12,35) = {1} The greatest element is 1.
D10 = {1, 2, 5, 10} D 75 = {1, 3, 5, 15, 25, 75} D (10,75) = {1,5} The greatest element is 5.
What can you say about the size of the sets Mk, Mh, and M[h,k]?
These sets are all infinite – that is they contain infinitely many elements.
What can you say about the size of the sets Dk, Dh, and D(h,k)
These sets are all finite – they contain only a finite number of elements.
What does it mean if D(h,k) = {1}? The two numbers h and k have no common factors other than 1.
If n and d are numerator and denominator of a fraction, what does it mean if D(n,d) = 1?
It means the fraction is reduced to lowest terms.
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