CEGEP CHAMPLAIN - ST. LAWRENCE
201-510-LW: Business Statistics
Patrice Camiré
Set Theory - Notes
Definition 1. A set is a collection of objects where order and repetition is irrelevant. If A is a set, we
let #A denote the number of elements in A. We say that x is an element of A, and write x ∈ A, if x is
contained in the set A.
Example 1.
1. The set A = {1, 3, 5, 9, 10, 13} contains 6 elements, so we write #A = 6.
2. 5 ∈ {1, 3, 5, 9, 10, 13}, but 7 6∈ {1, 3, 5, 9, 10, 13}.
3. Note that A = {1, 3, 5, 9, 10, 13} = {10, 5, 13, 9, 1, 3} = {1, 1, 1, 3, 5, 9, 9, 10, 13, 13}.
4. We let ∅ = { } denote the empty set, thus the set that contains no element.
Definition 2. Let A and B be sets. We say that A is a subset of B, and write A ⊆ B, if every element
of A is also an element of B. In other words, A is contained in B.
Example 2.
1. Let A = {2, 5, 6, 8} and B = {1, 2, 4, 5, 6, 7, 8, 9}. Then, A ⊆ B, so A is a subset of B.
2. Let A = {2, 3, 6, 8} and B = {1, 2, 4, 5, 6, 7, 8, 9}. Then, A 6⊆ B, so A is NOT a subset of B.
Definition 3. Let A and B be sets.
1. We denote and define the intersection of A and B by
A ∩ B = {x : x ∈ A and x ∈ B} .
In other words, A ∩ B is the set containing the elements common to both A and B.
We say that A and B are disjoint if A ∩ B = ∅.
2. We denote and define the union of A and B by
A ∪ B = {x : x ∈ A or x ∈ B} .
In other words, A ∪ B is the set obtained after combining all the elements of A and B together in
one larger set.
Example 3. Let A = {1, 2, 4, 5, 6, 7, 9} and B = {2, 3, 6, 8, 9}. Then,
1. A ∩ B = {2, 6, 9}.
2. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Definition 4. Let A and B be sets. We denote and define the set A minus B by
A − B = {x : x ∈ A and x 6∈ B}
In other words, we start with A and then proceed to remove from it the elements that are also in B.
Example 4. Let A = {2, 4, 6, 8, 10} and B = {1, 3, 4, 8, 9, 10}. Then A − B = {2, 6}.
Definition 5. We say that a set S is a universal set if it contains all the elements of interest in a particular
situation. If A ⊆ S, then we denote and define its complement by
Ac = {x : x ∈ S and x 6∈ A}
In other words, Ac contains all the elements in S that are not in A. Observe the following three key facts:
1. A ∩ Ac = ∅
2. A ∪ Ac = S
3. Combining the previous two facts yields #S = #A + #Ac
Definition 6. Let A and B be two sets. We denote and define their Cartesian product by
A × B = {(a, b) : a ∈ A and b ∈ B}
Observe that
#(A × B) = (#A)(#B)
Example 5. Let A = {1, 4, 6} and B = {0, 4, 7, 9}. Then
A × B = {(1, 0), (1, 4), (1, 7), (1, 9), (4, 0), (4, 4), (4, 7), (4, 9), (6, 0), (6, 4), (6, 7), (6, 9)}
Notice that #(A × B) = (3)(4) = (#A)(#B) = 12
Definition 7. A Venn diagram is a way of visualizing sets and their relationship to one another. You
should be able to figure it out with the corresponding problem sheet.
This idea may not seem like much now, but it will be surprisingly useful when we begin our discussion
of probability.
Theorem 1. Let A and B be sets. The following result is called the inclusion/exclusion principle:
#(A ∪ B) = #A + #B − #(A ∩ B)
You can naturally understand this result with the help of a Venn diagram.