The lesson introduces the important topic of sets, a simple idea that recurs throughout the study of probability and statistics.
Set Definitions
A set is a well-defined collection of objects.
Each object in a set is called an element of the set.
Two sets are equal if they have exactly the same elements in them.
A set that contains no elements is called a null set or an empty set.
If every element in Set A is also in Set B, then Set A is a subset of Set B.
Set Notation
A set is usually denoted by a capital letter, such as A, B, or C.
An element of a set is usually denoted by a small letter, such as x, y, or z.
A set may be described by listing all of its elements enclosed in braces. For example, if Set Aconsists of the numbers 2, 4, 6, and 8, we may say: A = {2, 4, 6, 8}.
The null set is denoted by {∅}.
Sets may also be described by stating a rule. We could describe Set A from the previous example by stating: Set A consists of all the even single-digit positive integers.
Set Operations
Suppose we have four sets - W, X, Y, and Z. Let these sets be defined as follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}.
The union of two sets is the set of elements that belong to one or both of the two sets. Thus, set Z is the union of sets X and Y.
Symbolically, the union of X and Y is denoted by X ∪ Y.
The intersection of two sets is the set of elements that are common to both sets. Thus, set W is the intersection of sets X and Y.
Symbolically, the intersection of X and Y is denoted by X ∩ Y.