If A and B are sets with the property that every element of A is an element of B, then A is a subset of B. For example,
The union of two sets is the set consisting of everything which is an element of at least one of the sets, A or B. As an example of the union of two sets
Be sure you understand that something which is in both A and B is in the union. It is not an exclusive or.
The intersection of two sets, A and B consists of everything which is in both of the sets. Thus
A special set which needs to be given a name is the empty set also called the null set, denoted by ∅. Thus ∅ is defined as the set which has no elements in it. Mathematicians like to say the empty set is a subset of every set. The reason they say this is that if it were not so, there would have to exist a set A, such that ∅ has something in it which is not in A. However, ∅ has nothing in it and so the least intellectual discomfort is achieved by saying ∅⊆ A.
If A and B are two sets, A ∖ B denotes the set of things which are in A but not in B. Thus
Set notation is used whenever convenient.
To illustrate the use of this notation relative to intervals consider three examples of inequalities. Their solutions will be written in the notation just described.
Example 1.1.1 Solve the inequality 2x + 4 ≤ x − 8
x ≤−12 is the answer. This is written in terms of an interval as (−∞,−12].
Example 1.1.2 Solve the inequality
The solution is x ≤−1 or x ≥
Example 1.1.3 Solve the inequality x
This is true for any value of x. It is written as ℝ or
Something is in the Cartesian product of a set whose elements are sets if it consists of a single thing taken from each set in the family. Thus
signifies the Cartesian product.
The Cartesian product is the set of choice functions, a choice function being a function which selects exactly one element of each set of S. You may think the axiom of choice, stating that the Cartesian product of a nonempty family of nonempty sets is nonempty, is innocuous but there was a time when many mathematicians were ready to throw it out because it implies things which are very hard to believe, things which never happen without the axiom of choice.