Download TC Worksheet #2

Sets
A set is a collection of objects, called the elements of the set. The order in which the elements of a set are listed
is not relevant; the set is the same regardless of the order in which the elements are listed.
Sets can be represented in a number of ways including the following:
 Roster notation: Using roster notation, the elements of the set are separated by commas and enclosed in
braces, { }. No element is listed more than once. For example, the set of days of the week that begin
with the letter S is {Saturday, Sunday}. Because the order in which the elements are listed is not
important, the set could also be written as {Sunday, Saturday}.
If there are too many elements to list, then an ellipsis is used to indicate that the elements continue in the
same pattern. For example, the set of even numbers between 0 and 100 could be written as
{2, 4, 6, 8, …, 98}, and the set of all positive even numbers could be written as {2, 4, 6, 8, 10, …}.
It is important that the set be enclosed in braces and that the elements be separated by commas. The set
{cat} contains only one element, the word cat, whereas the set {c, a, t} contains three elements, the
letters c, a, and t.

Set-builder notation: Using set-builder notation, a variable is used to represent an arbitrary element of
the set. All values which satisfy the description of the variable are elements of the set.
For example, you could use set-builder notation to define the set {Saturday, Sunday} by letting x be the
variable and using the description x is a day of the week that begins with the letter S. The set would be
typed as {x | x is a day of the week that begins with the letter S}, which is read as, “the set of x such that
x is a day of the week that begins with the letter S.”
Note: You have the symbols {, }, and | on your keyboard.
Some special sets of numbers have names and notations that are commonly used, for example:
 N = the set of natural numbers = {1, 2, 3, 4, 5, …}
 Z = the set of integers = {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}
 E = the set of even natural numbers = {2, 4, 6, 8, 10, …}
 O = the set of odd natural numbers = {1, 3, 5, 7, 9, …}
How to type set operations
You will be introduced to several set operations, but the symbols for these are not available on the standard
computer keyboard. One option is to copy and paste the symbols into the answer box in the homework or
exam. Another option is to use the following alternative symbols. If you are unsure how to type a symbol into
the computer, please ask your instructor before submitting your assignment.



Intersection: type “n” in place of ∩
Example: A ∩ B would be typed as A n B
Empty set: type { } instead of ∅
Is an element of: type “(-” in place of ∈
Example: x ∈ A would be typed as x (- A
Sets
Examples
(Solutions provided at bottom of page)
Represent each set described using the roster method.
1. The letters in the word banana
2. The days of the week that begin with the letter T
3. The months of the year that end in the letter y
4. The seasons of the year
5. The integers between 1 and 10
6. The letters that the words Canada and France have in common
Represent each set using both roster and set-builder notation.
7. The set of odd numbers between 10 and 100
8. The set of even numbers greater than 3
9. The set of integers less than 0
10. The set of natural numbers greater than 100
List the smallest three values in each set.
11. {x | x ∈ N and x > 5}
12. {x | x ∈ E and 10 < x < 30}
13. {x | x ∈ O and x > 100}
Determine whether or not each number is in the set {x | x ∈ Z and x < 5}.
14. x = -3
15. x = 2/3
Solutions:
1. {b, a, n}
2. {Tuesday, Thursday}
3. {January, February, May, July}
4. {Winter, Spring, Summer, Fall}
5. {2, 3, 4, 5, 6, 7, 8, 9}
6. {a, n}
7. {11, 13, 15, 17, …, 99}; {x | x ∈ O and 10 < x < 100} or {x | x is odd and 10 < x < 100}
8. {4, 6, 8, 10, 12, 14, …}; {x | x ∈ E and x > 3} or {x | x is even and x > 3}
9. {-1, -2, -3, -4, -5, …}; {x | x ∈ Z and x < 0} or {x | x is an integer and x < 0}
10. {101, 102, 103, 104, 105, …}; {x | x ∈ N and x > 100} or {x | x is a natural number and x > 100}
11. 5, 6, 7
12. 12, 14, 16
13. 101, 103, 105
14. Yes
15. No
Worksheet 2
Sets
Represent each set described using the roster method.
1. The letters in the word alfalfa
2. States that begin with the letter C
3. Months that end with the letter r
4. Even integers between 10 and 20
5. The letters that the words history and biology have in common
6. The letters in the alphabet that are always vowels
List the smallest three values in each set.
7. {x | x ∈ N and 20 < x}
8. {x | x ∈ E and x < 30}
9. {x | x ∈ Z and 100 < x < 130}
Determine whether or not each number is in the set {x | x ∈ O and x < 5}.
10. a.
b.
c.
d.
e.
x = -3
x=0
x=1
x=3
x=5