A Probability Course for the Actuaries A Preparation for Exam P/1

14 BASIC OPERATIONS ON SETS
Solution.
(a) n(∅) = 0.
(b) This is a set consisting of one element ∅. Thus, n({∅}) = 1.
(c) n({a, {a}, {a, {a}}}) = 3
Now, one compares numbers using inequalities. The corresponding notion
for sets is the concept of a subset: Let A and B be two sets. We say that
A is a subset of B, denoted by A ⊆ B, if and only if every element of A is
also an element of B. If there exists an element of A which is not in B then
we write A ⊈ B.
For any set A we have ∅ ⊆ A ⊆ A. That is, every set has at least two subsets.
Also, keep in mind that the empty set is a subset of any set.
Example 1.6
Suppose that A = {2, 4, 6}, B = {2, 6}, and C = {4, 6}. Determine which of
these sets are subsets of which other of these sets.
Solution.
B ⊆ A and C ⊆ A
If sets A and B are represented as regions in the plane, relationships between
A and B can be represented by pictures, called Venn diagrams.
Example 1.7
Represent A ⊆ B ⊆ C using Venn diagram.
Solution.
The Venn diagram is given in Figure 1.1
Figure 1.1