Sets
Basic Definitions
Set: Unordered collection of objects
Objects: elements or members of a set. A set contains elements.
Sets can contain anything.
Notation
There are a couple of ways to describe a set. You can list all the elemnts or objects in the set. {w, x, y} is a set with three elements in it. Notice the brackets {} are used to contain a set.
Ex.1
V is a set containing all vowels in the English language.
V = {a, e, i, o, u}
O is a set containing odd positive integers less than 10.
O = {1, 3, 5, 7, 9}
Usually the elements in a set will have common properties. However, a set can contain unrelated elements. {x, 3, Jer, Canada} is still an acceptable set.
Another notation to describe a set is using a Set Builder.The elements in a set are described using a property or properties the objects must have to be members.
Ex.2
O is a set containing odd positive integers less than 10
O = {x | x is an odd positive integer less than 10}
Read the last statement like this.
The set 'O' contains elements 'x' such that 'x' is an odd positive integer less than 10.
Notice the '=' sign means 'contains'. The '|' sign means 'such that'.
So the elements in O are {1,3,5,7,9}.
Check out the notation page for some important sets
A Little More Advanced Definitions
= Equal: Two sets are equal if and only if they contain the same elements.
So if A and B are sets, then A and B are equal if and only if,
∀x (x ∈ A ↔ x ∈ B)
If A and B are equal, we write A = B.
Order does not matter is a set.
Sets {3, 5, 7} and {5, 7, 3} are equal.
Repeated elements in a set do not matter.
Sets {3, 3, 5, 5, 5, 7} and {3, 5, 7} are equal.
ø {} Empty Set or Null Set: A set with no elements.
Singleton Set: A set with one element.
A singleton set could contain the set {ø}. This is a singleton set containing the empty set.
Note the difference between the empty set ø and the set {ø}.
The set {ø} is a singleton set since it contains the element ø. An analogy to for this having a bank account with no money in it {ø}, is different from not having a bank account ø.