Basic Set Theory

Venn diagram of set intersection
What Is a Set?

Informally, a set is a collection of objects. Often, these are mathematical objects, but they need not be. The populations studied in statistics can be regarded as sets.
No object can be included in a set multiple times.
Examples:

Set membership

We can define a set by listing its members. The notation is:
A = {1, 2, 3, 4, 5, 6}
This might be the set of possible rolls of a six-sided die.

Or we can use a rule based definition. If we want all even integers, we could write:
A = {x: x is even}

Subsets



Let us define A as "the set of all states in the U.S."
Then we might define two more sets as R = "the set of states that voted for Romney in 2012," and O = "the set of all states that voted for Obama in 2012."



Then R and O are both subsets of A. We define a subset of a set A as any set that contains only members A.
Our notation here is:
R ⊆ A
O ⊆ A

Note that under this definition, A is a subset of itself. We also can define proper subsets of A: these contain only members of A but not all members of A. Both R and O are also proper subsets of A. Our notation is:
R ⊂ A
O ⊂ A

Some special sets
Operations on Sets
Set equality

We say two sets are equal if they contain the same members. So, if A = {1, 2, 3} and B = {3, 2, 1}, then we can write A = B. (Order does not matter!)

Union of sets

The union operator combines two sets so that the new set contains all of the elements of each of the two.



If today's baseball game is the Angels versus the Yankees, then the "set of all players" (P) is the union of the "set of all Angels players" (A) and the "set of all Yankees players" (Y).
Our notation here is:
P = A ∪ Y

Intersection of sets

The intersection of two sets contains only the elements that are in both sets. So if R = {the states that have voted for a Republican candidate for president since 2000} and D = {the states that have voted for a Democratic candidate for president since 2000} then their intersection is the states that are not pure red or pure blue in this map:



This would be the set of swing states, S.
Our notation here is:
S = R ∩ D

Complement of a set

We can define the universe of some group of sets as the collection of all possible members. So if we were surveying American voters, then U = {all American voters}.

Or if we are considering American states, then U = {all American states}.

Then, if O = Obama states, and R = Romney states, those sets are complements. We have:
O ∪ R = U
O ∩ R = ∅

Venn Diagrams


Basics

These diagrams are a great way to get a picture of the ideas of set theory. We depict a set as a circle.

Then, for the intersection of two sets, we have the diagram:


Or for the union of two sets, we have the diagram:


Finally, if A is the set in the left circle below, the complement of A is:

De Morgan's laws
(Not covered Summer 2017.)



In English: the bar over the symbols means essentially "not." And what is comprised by each equation is the area in blue.

So:
Law 1: what is not in the union of two sets is the intersection of what is not in one of them and what is not in the other of them.
Example: U (our universe) = "people authorized to be on our campus"
"The people authorized to be on our campus who are not either students at St. Joseph's College or faculty at St. Joseph's College" (a union of two sets) is equivalent to "The people authorized to be on our campus who are not on the list of students and who are not on the list of faculty" (an intersection of two sets). This set would include administrators, workers at the cafeteria, visitors with passes, security personnel, and so on.

Law 2: what is not in the intersection of two sets is the union of what is not in one of them and what is not in the other of them.
Example: U (our universe) = "people authorized to be on our campus"
"The people authorized to be on our campus who are not both students at St. Joseph's College and faculty at St. Joseph's College" (an intersection of two sets) is equivalent to "The people authorized to be on our campus who are not students or who are not faculty" (a union of two sets). This set contains everyone authorized on campus who is not a student and a faculty member. (Last semester, that set might have excluded only me!)

External Links