1. Review of Sets and Set Operations
Chapter 1-1
Review of Sets and Set Operations

2. What’s a Set?
A set is a collection whose members are specified by a list or rule
In the list, only have each element listed once (no need to have a duplicate element in a set)
Ex: 𝑆= 𝐴𝑙𝑎𝑏𝑎𝑚𝑎, 𝐴𝑙𝑎𝑠𝑘𝑎, 𝐴𝑟𝑖𝑧𝑜𝑛𝑎, 𝐴𝑟𝑘𝑎𝑛𝑠𝑎𝑠
Set S is the set of states starting with the letter A
When a rule is used to specify a set you will see it written as:
𝑆= 𝑥:… This is read “S is a set of all x such that…
Nothing can be partially in a set. It is either in the set, or it isn’t.
Finite sets have a finite number of elements within it
Ex: 𝑌= 𝑦:𝑦 𝑖𝑠 𝑎𝑙𝑙 𝑒𝑣𝑒𝑛 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10
Infinite sets have infinite number of elements within it
Ex: Z= 𝑧:𝑧 𝑖𝑠 𝑎𝑙𝑙 𝑜𝑑𝑑 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

3. Elements
The set I of even positive integers less than 15 can be written a couple ways
Ex: 𝐼= 𝑛:𝑛 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 15
Ex: 𝐼= 2,4,6,8,10,12,14
To indicate x is an element of a set X, we write 𝑥∈𝑋
To indicate x is not an element of set X, we write 𝑥∈𝑋
Ex: 2∈𝐼 “2 is an element of set I”
**important note: lowercase letters indicate the element, uppercase letters indicate the set**

4. Subsets
A set A is a subset of a set B, written 𝐴 𝐵, if every element in A is also in B.
If A B and B A, then B and A have exactly the same elements, in which case we say 𝐴=𝐵
Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14 𝐼= 2,4,6,8,10,12,14
Since all elements in A are also in I, A I
Since all elements in B are also in I, B I
Is A B?
Is B A?

5. Unions A Union is a combination of all the elements of 2 sets
Let A and B be sets. The set 𝐴∪𝐵, called the union of A and B, consists of all elements which are in A or B or both.
𝐴∪𝐵= 𝑥:𝑥∈𝐴 𝑜𝑟 𝑥∈𝐵
Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14
𝐴∪𝐵= 2,4,6,8,10,12,14
Even though 2 is in both sets, we do not need to write 2 twice in the Union set

6. Intersections Intersections are like Unions, but opposite.
An intersection of sets contains only elements that are in common in both sets
Let A and B be sets. The set 𝐴∩𝐵, is called the intersection of A and B, consists of all elements which are in both A and B
𝐴∩𝐵= 𝑥:𝑥∈𝐴 𝑎𝑛𝑑 𝑥∈𝐵
Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14
𝐴∩𝐵= 2, and 12 are the only values in both sets

7. Try this Let sets S, E, C, and M be defined as
𝑆= CT, MA, MD, CA, CO, MI, MN
𝐸= CT,MA,MD
𝐶= CA,CO,CT
𝑀= MA,MD, MI,MN
Name any subsets you see.
𝐸∩𝐶=
𝐸∩𝑀=
𝐸∪𝐶=
𝐸∪𝑀=
𝑀∪𝐶=
𝑀∩𝐶=

8. Empty Set
The set which contains no elements is known as the empty set, and is denoted by ∅. By convention, the empty set is a subset of every set
Since the empty set has no elements, we say 𝐴∩∅=∅ and A∪∅=𝐴
Two sets A and B are disjoint if 𝐴∩𝐵=∅
This means sets A and B have no elements in common

9. Parentheses Matter! Let 𝐴= 𝑎,𝑏,𝑐 𝐵= 𝑎,𝑐,𝑒 𝐶= 𝑎,𝑑 Ex: 𝐴∩𝐵∩𝐶= Ex: 𝐴∪𝐵∪𝐶=
Notice:

10. Universal Set and Complements
A set U is said to be a universal set for a problem if all sets being considered in the problem are subsets of U. Let A be subset of U, then A’ is the complement of A, meaning A’ contains all the elements that are not in A.
A’ is said “A Prime”
Ex: Let 𝑈= CA,CO,CT,IL,IN 𝑋= CA,CT,IL 𝑌= CO,CT, IN 𝑍= CO,IN
𝑋 ′ =
𝑌 ′ =
𝑍 ′ =
𝑌∩ 𝑍 ′ =
𝑋∩ 𝑍 ′ =
𝑍∩ 𝑌 ′ =

11. Cartesian Product
The Cartesian Product of sets A and B, denoted 𝐴×𝐵, is the set of all ordered pairs 𝑎,𝑏 where 𝑎∈𝐴 and 𝑏∈𝐵
𝐴×𝐵= 𝑎,𝑏 :𝑎∈𝐴, 𝑏∈𝐵
Ex: A survey can be conducted by either mail (M) or phone (P) in one of three cities: Atlanta (A), Boston (B), or Cincinnati (C). You must first choose a method (M, P) then a city (A, B, C). Each possible survey can be denoted as an ordered pair.
𝑀,𝐴 , 𝑀,𝐵 , 𝑀,𝐶 , 𝑃,𝐴 , 𝑃,𝐵 ,(𝑃,𝐶)

12. Cartesian Product (You Try)
Let 𝐴= 𝑎,𝑐,𝑒 𝐵= 𝑏,𝑑,𝑒 𝐶= 𝑏,𝑑 \
Find 𝐴×𝐶=
Find 𝐵×𝐶=
Find 𝐶×𝐶=
The order of elements within the braces is not important but the order of the symbols within the parentheses is! (𝑥,𝑦) is very different from (𝑦,𝑥)

13. Order Matters!
A football league consists of 4 teams: Aardvarks (A), Bisons (B), Coyotes (C), and Dingos (D). Each game can be denoted as an ordered pair in which the first entry denotes the home team.
Games: 𝐺=