Introduction to Sets
Some Exercises
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}
b) { x | x is an integer such that x² = 2}
c) { x | x is a positive integer such that x < 2}
d) { x | x is a positive integer such that x < 2 or 4 < x < 7}
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}
b) {{1}}, {1, {1}}
c)
,{ }
3. Write each of the following sets with set builder notation. Sketch their graphs on a Real Number line.
a) {1,2,3,4}
b) {2, 5, 8, 11, 14}
c) {-3, -2, -1, 0, 1, 2, 3}
d) (-4, 8)
e) (3, 10]
f) [-14, 2)
g) [0, 3]
h) (-2, 0)(0,1)
i) (-2, 0) (0,1)
j) [-2, 0](0,1]
k) (-∞, 5)(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}
b) {x| x  N and -3 < x < -1}
c) {x| x  Z and -3 < x < -1}
d) {x| x  W and -3 < x < -1}
e) {x| x  Z and -3 < x < -1 or x  N}
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}
b) {x| x  R and (x < 8 or x > 12)}={ x  R | x < 8 or x > 12}
c) {x| x  R and 3 < x < 8} = {x  R|3 < x and x < 8}= { x  R|3 < x < 8}
d) {x  R | x < -3 or x > 1}
e) {x  R | x < 5 or x > 0}
f) {x  R | x > 5 or x > 0}
g) { x  R | x < 5 or x < 0}
h) {x  R|3 < x and x > 0}
i) {x  R|1 < x and x < 3}
j) {x  R|-2 < x and x < 1}
k) {x  R|3 > x and x < 8}
l) {x  Q|-2 < x and x < 1} F
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6. 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}
a) Write all of the subset relations that exist between A, B, C, and D.
b) Compute A  B. Draw a Venn diagram to illustrate.
c) Compute A  B Draw a Venn diagram to illustrate.
d) Compute B  C  D Draw a Venn diagram to illustrate.
e) Compute B  C D Draw a Venn diagram to illustrate.
f) Compute B  C Draw a Venn diagram to illustrate.
g) Compute D  (B  C) and compute (D  B)  (D  C)
h) Compute C  
i) Compute D  
7. 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)
b) N  (-2,0]
c) N  [3,5]
d) Z  (2,5)
e) Z  (-2,0]
f) Z  [3,5]
g) R  (2,5)
h) R  (-2,0]
i) R  [3,5]
j) Q  (2,5)
k) Q  (-2,0]
l) Q  [3,5]
For Problems 8 – 12 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}.
8. 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.
9. 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.
10. What is A B ? How is it related to B? How is it related to A?
11. What is B C ? How is it related to B? How is it related to C?
12. What is A B ? How is it related to A? How is it related to 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.
Exercise 2: Use the extended roster method to write the set of multiples of 5. Call this set M5.
Exercise 3: Use the extended roster method to write M3M5. Call this set M[3,5].
Exercise 4: How would you describe the elements of M[3,5]? Give two descriptions.
Exercise 5: How would you name the smallest element of M[3,5]?
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
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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
b) M3, M8, M[3,8] ,and its least element
c) M6, M18, M[6,18] ,and its least element
d) M10, M14, M[10,14] ,and its least element
and so on – that is called studying.
Exercise 6: Use the roster method to write the set of divisors of 24. Call this set D24.
Exercise 7: Use the roster method to write the set of divisors of 15. Call this set D15.
Exercise 8: Use the roster method to write D24D15. Call this set D(24,15).
Exercise 9: How would you describe the elements of D(24,15)?
Exercise 10: How would you name the largest element of D(24,15)?
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
and so on – that is called studying.
What can you say about the size of the sets Mk, Mh, and M[h,k]?
What can you say about the size of the sets Dk, Dh, and D(h,k)
What does it mean if D(h,k) = {1}?
If n and d are numerator and denominator of a fraction, what does it mean if D(n,d) = 1?
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