Week 6: Sets
A set is a Many that allows itself to be thought of as a One.
Georg Cantor
Basic Ideas
A set is a collection of objects, often called the elements or members of the set. When we
give a set a name, it is usually a capital letter like “𝐴,” “𝐵,” etc. If the set is used very
frequently, like the sets of numbers we have seen in the past, it is given its own special
symbol that normally resembles a capital letter. This naming scheme is just one of the
many conventions of mathematics; there is no “rule” that says we can’t use the letter 𝑥 to
represent the set of all even integers, for example, but it would be confusing to experienced
readers.
As we have already seen, we use the symbol “∈” to indicate that an object belongs to a set.
For example, we can say that −1 ∈ ℤ and −1 ∈ ℚ, but −1 ∉ ℕ. Similarly, √2 ∈ ℝ but
√2 ∉ ℚ.
There are two main notations for describing a particular set. One of these is to explicitly list
the elements of the set inside “curly braces.” For example, think about the set of all even
natural numbers less than 25. If we call this set 𝐸, we can describe it as follows:
𝐸 = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.
Alternatively, we could write 𝐸 = {0, 2, 4, 6, … , 24}. Using an ellipsis like this relies on the
reader’s ability to follow the pattern shown by the first few elements listed. Similarly, we
could describe the infinite sets of natural numbers and integers as follows:
ℕ = {0, 1, 2, 3, 4, … } and
ℤ = {… , −3, −2, −1, 0, 1, 2, 3, … }.
A second, more versatile way to describe a set is often called “set builder notation.” In this
notation, the set 𝐸 given above would be written as follows:
𝐸 = {𝑥 ∈ ℕ ∶ 𝑥 is even and 𝑥 < 25}.
The common way to read this statement is “𝐸 is the set containing each natural number 𝑥
such that 𝑥 is even and 𝑥 < 25.” As another example, we could describe the rationals in this
way:
ℚ = {𝑎/𝑏 ∶ 𝑎 ∈ ℤ and 𝑏 ∈ ℤ and 𝑏 ≠ 0}
This is “the set of all fractions 𝑎/𝑏 such that 𝑎 and 𝑏 are integers and 𝑏 is not 0.” The colon
in set builder notation roughly stands for the words “such that.” Some authors use a
vertical line “|” in place of a colon.
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We use the symbol ∅ to represent a set with no elements, which is called the empty set, or
sometimes the “null set.” In other words, ∅ = { }.
Operations
Just as we can combine numbers in various ways to make other numbers, or combine
logical statements in various ways to formulate new statements, we can combine sets to
construct new sets.
Definition: The union of sets 𝐴 and 𝐵, which is written as 𝐴 ∪ 𝐵, is given by
𝐴 ∪ 𝐵 = {𝑥 ∶ 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵}.
Definition: The intersection of sets 𝐴 and 𝐵, which is written as 𝐴 ∩ 𝐵, is given by
𝐴 ∩ 𝐵 = {𝑥 ∶ 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵}.
Definition: The difference of sets 𝐴 and 𝐵, which is written as 𝐴 − 𝐵 (or sometimes 𝐴\𝐵),
is given by
𝐴 − 𝐵 = {𝑥 ∶ 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐵}.
Definition: The complement of a set 𝐴, which is written as 𝐴𝐶 (or sometimes 𝐴′ or 𝐴̃ or 𝐴̅),
is given by
𝐴𝐶 = {𝑥 ∶ 𝑥 ∉ 𝐴}.
To illustrate these definitions, consider the set 𝐸 = {𝑥 ∈ ℕ ∶ 𝑥 is even and 𝑥 < 25} from
before, and a new set 𝑆 = {𝑥 ∈ ℤ ∶ |𝑥| ≤ 5}. Observe the following:
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𝐸 ∪ 𝑆 = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.
𝐸 ∩ 𝑆 = {0, 2, 4}.
𝐸 − 𝑆 = {6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.
𝑆 − 𝐸 = {−5, −4, −3, −2, −1, 1, 3, 5}.
The complement of a set is the collection of all objects that do not belong to the original set.
The exact meaning of the complement depends on what kinds of things we consider to be
“objects.” The set of all possible “objects” in a given context is usually known as the
universe, or the universe of discourse. For example, if we are working in an area of
mathematics that only involves integers, we might say that the complement of the set 𝑆 (as
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given above) is the set of all integers less than −5 and all integers greater than 5. In that
same universe of discourse, ℕ𝐶 would be the set of all negative integers (also known as ℤ− ).
Similarly, if we are working in an area that deals with all real numbers, the set of irrationals
could be abbreviated as ℚ𝐶 . This is perhaps part of the reason that there is no widely-used
symbol for the set of irrational numbers.
Exercise 1: For the sets 𝐴 = {1,2,3,4,5}, 𝐵 = {2,4,6,8,10}, and 𝐶 = {8,10,12}, find each of
the following.
(a) 𝐴 ∩ 𝐵
(b) 𝐴 ∩ 𝐵 𝐶
(c) 𝐴 ∪ 𝐵
(d) 𝐴 ∩ 𝐶
(e) 𝐴 − 𝐵
(f) 𝐵 − 𝐴
Exercise 2: Give a very brief abbreviation for the set of irrationals without using a
complement symbol.
It is worth pointing out here that in advanced areas of mathematics, we often need to think
about the union of more than two sets, or the intersection of more than two sets. In some
cases, we think about the union or intersection of an infinite number of sets. Naturally, the
union of a given collection of sets is a set containing every object that is an element of at
least one of the sets in the collection. Similarly, the intersection of a given collection of sets
is a set containing every object that is an element of every set in the collection.
As an example, think about all of the subsets of ℕ with at least five elements. The union of
these subsets would be ℕ itself, because every element of ℕ is an element of at least one of
these subsets. On the other hand, the intersection of these subsets would be ∅, because
there is no number that is an element of every one of the subsets.
Statements
One of the most common ways in which two sets might be related to each other is for one
of them to be a subset of another. When all of the elements of one set belong to a second set
as well, we say that the first set is a subset of the other.
Definition: We say that a set 𝐴 is a subset of a set 𝐵, which is written as 𝐴 ⊆ 𝐵, iff
(∀𝑥 ∈ 𝐴)(𝑥 ∈ 𝐵).
Note that “𝐴 ⊆ 𝐵” means that every element of 𝐴 is an element of 𝐵. If every element of 𝐵 is
also an element of 𝐴, then the two sets have exactly the same elements, and therefore they
are the same set. It is helpful to state this observation as an official definition.
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Definition: A set 𝐴 is equal to a set 𝐵, which is written as 𝐴 = 𝐵, iff 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴.
When one set is a subset of another, but we know that they are not the same set, we can say
that the first set is a “proper” subset of the second.
Definition: We say that a set 𝐴 is a proper subset of a set 𝐵, which is written as 𝐴 ⊂ 𝐵, iff
𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵.
Unfortunately, it must be noted here that some authors use the symbol “⊂” to represent
any subset relationship, whether it is proper or not. To avoid confusion, we will never do
this!
Lastly, we often deal with situations in which two sets have nothing in common
whatsoever.
Definition: Sets 𝐴 and 𝐵 are disjoint iff 𝐴 ∩ 𝐵 = ∅.
Exercise 3: Consider the sets 𝐴 = {𝑥 ∶ 𝑥 = 5𝑘 + 1 for some natural number 𝑘} and 𝐵 =
{𝑥 ∶ 𝑥 = 6𝑚 for some natural number 𝑚}. Prove that each of the following statements is
false.
(a) 𝐴 ⊆ 𝐵
(b) 𝐵 ⊆ 𝐴
(c) 𝐴 ∩ 𝐵 = ∅
Exercise 4: Disprove each of the following general statements about sets.
(a) For any sets 𝐴 and 𝐵, if 𝐴 ⊆ 𝐵, then 𝐴 − 𝐵 = 𝐵 − 𝐴.
(b) For any sets 𝐴, 𝐵, and 𝐶, (𝐴 − 𝐵) ∪ (𝐵 − 𝐶) ⊆ (𝐴 − 𝐶).
(c) For any sets 𝐴, 𝐵, and 𝐶, 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ 𝐶.
Proofs
Keep in mind that a set can contain any kind of object; they are not restricted to numbers.
All of the specific sets we have seen so far have been sets of numbers, and there are some
simple relationships between these sets – for example, ℤ+ ⊆ ℕ, and ℕ ⊆ ℤ, etc. However, it
is more interesting to prove some general properties of sets. In other words, there are
statements about sets that are true regardless of what the sets might specifically contain.
Following are some examples.
Proposition: For any sets 𝐴, 𝐵, and 𝐶, if 𝐴 ⊆ 𝐵, then 𝐴 ∩ 𝐶 ⊆ 𝐵 ∩ 𝐶.
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Proof: Let 𝐴, 𝐵, and 𝐶 be sets and 𝐴 ⊆ 𝐵. Now, we must prove that 𝐴 ∩ 𝐶 is a subset of
𝐵 ∩ 𝐶. Based on the definition of “subset”, this requires that we prove that every element of
𝐴 ∩ 𝐶 must be an element of 𝐵 ∩ 𝐶. Let 𝑥 be any element of 𝐴 ∩ 𝐶. By definition, 𝑥 is an
element of both 𝐴 and 𝐶. Since 𝑥 is an element of 𝐴 and 𝐴 ⊆ 𝐵, 𝑥 must be an element of 𝐵.
Since we now know 𝑥 is an element of both 𝐵 and 𝐶, by definition 𝑥 is an element of
𝐵 ∩ 𝐶. Therefore 𝐴 ∩ 𝐶 ⊆ 𝐵 ∩ 𝐶. ∎
Proposition: For any sets 𝐴, 𝐵, and 𝐶, 𝐴 ⊆ 𝐵 if and only if 𝐴 ∩ 𝐵 = 𝐴.
Proof: Let 𝐴, 𝐵, and 𝐶 be sets. Remember that proving a biconditional statement requires
two proofs. We will first prove that if 𝐴 ⊆ 𝐵, then 𝐴 ∩ 𝐵 = 𝐴. Suppose 𝐴 ⊆ 𝐵. We must
prove that 𝐴 ∩ 𝐵 ⊆ 𝐴 and 𝐴 ⊆ 𝐴 ∩ 𝐵. First, suppose 𝑥 is any element of 𝐴 ∩ 𝐵. By definition,
𝑥 is an element of 𝐴 (and also of 𝐵), and therefore 𝐴 ∩ 𝐵 ⊆ 𝐴. Now, suppose 𝑦 is any
element of 𝐴. Since we have supposed 𝐴 ⊆ 𝐵, by definition 𝑦 must be an element of 𝐵. Since
𝑦 is an element of both 𝐴 and 𝐵, 𝑦 is an element of 𝐴 ∩ 𝐵 by definition, and therefore
𝐴 ⊆ 𝐴 ∩ 𝐵.
We now prove that if 𝐴 ∩ 𝐵 = 𝐴, then 𝐴 ⊆ 𝐵. Suppose 𝐴 ∩ 𝐵 = 𝐴. We must prove that
𝐴 ⊆ 𝐵. So let 𝑧 be any element of 𝐴. Since 𝐴 = 𝐴 ∩ 𝐵, 𝑧 must be an element of 𝐴 ∩ 𝐵. Then
by definition, 𝑧 is an element of 𝐵, so 𝐴 ⊆ 𝐵. ∎
Proposition: For any sets 𝐴, 𝐵, 𝐶, and 𝐷, if 𝐶 ⊆ 𝐴 and 𝐷 ⊆ 𝐵, and 𝐴 and 𝐵 are disjoint, then
𝐶 and 𝐷 are disjoint.
Proof: Let 𝐴, 𝐵, 𝐶, and 𝐷 be sets and suppose 𝐶 ⊆ 𝐴, 𝐷 ⊆ 𝐵, and 𝐴 ∩ 𝐵 = ∅. We must prove
that 𝐶 ∩ 𝐷 = ∅. By definition, we must prove that 𝐶 ∩ 𝐷 has no elements. Suppose there is
an element 𝑥 in 𝐶 ∩ 𝐷. By definition, 𝑥 is an element of both 𝐶 and 𝐷. Then since 𝐶 ⊆ 𝐴 and
𝐷 ⊆ 𝐵, 𝑥 is an element of 𝐴 and of 𝐵, so 𝑥 is an element of 𝐴 ∩ 𝐵, which is ∅. This
contradicts the definition of ∅, so 𝐶 ∩ 𝐷 must be empty. ∎
In Exercises 5 – 13, prove each of the general properties about sets 𝐴, 𝐵, and 𝐶. Use only
definitions in your proofs; do not use previously-proved properties in new proofs.
Exercise 5: If 𝐴 ⊆ 𝐶 and 𝐵 ⊆ 𝐶, then 𝐴 ∪ 𝐵 ⊆ 𝐶.
Exercise 6: 𝐴 ∩ 𝐴𝐶 = ∅.
Exercise 7: (𝐴 ∪ 𝐵)𝐶 = 𝐴𝐶 ∩ 𝐵 𝐶 .
Exercise 8: If 𝐴 ∩ 𝐵 = 𝐴 and 𝐵 ∩ 𝐶 = 𝐵, then 𝐴 ∩ 𝐶 = 𝐴.
Exercise 9: 𝐴 ∪ (𝐴𝐶 ∩ 𝐵) = 𝐴 ∪ 𝐵.
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Exercise 10: 𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶).
Exercise 11: If 𝐴 ∩ 𝐵 = 𝐴, then 𝐴 ∪ 𝐵 = 𝐵.
Exercise 12: If 𝐴 ⊆ 𝐵, then 𝐴 ∩ 𝐵 𝐶 = ∅.
Exercise 13: (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶) = 𝐴 ∪ (𝐵 ∩ 𝐶).
Final Note: Interval Notation
Because much of mathematics – especially applied mathematics – deals with the set of real
numbers, some specialized notation has been developed for certain kinds of subsets of ℝ,
which we call “intervals.”
Definitions: Let 𝑎 and 𝑏 be any real numbers with 𝑎 ≤ 𝑏. The following definitions are
collectively called interval notation.
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(𝑎, 𝑏) represents the open interval {𝑥 ∈ ℝ ∶ 𝑎 < 𝑥 < 𝑏}.
[𝑎, 𝑏] represents the closed interval {𝑥 ∈ ℝ ∶ 𝑎 ≤ 𝑥 ≤ 𝑏}.
(𝑎, 𝑏] represents the set {𝑥 ∈ ℝ ∶ 𝑎 < 𝑥 ≤ 𝑏}.
[𝑎, 𝑏) represents the set {𝑥 ∈ ℝ ∶ 𝑎 ≤ 𝑥 < 𝑏}.
(−∞, 𝑏) represents the set {𝑥 ∈ ℝ ∶ 𝑥 < 𝑏}.
(𝑎, ∞) represents the set {𝑥 ∈ ℝ ∶ 𝑥 > 𝑎}.
(−∞, 𝑏] represents the set {𝑥 ∈ ℝ ∶ 𝑥 ≤ 𝑏}.
[𝑎, ∞) represents the set {𝑥 ∈ ℝ ∶ 𝑥 ≥ 𝑎}.
Exercise 14: Think about all the open intervals of the form (0, 1 + 𝑐), with 𝑐 being any
positive real number. For example, some of these intervals are (0, 2), (0, 100), (0, 2𝜋),
10
(0, 9 ), and so on.
(a) What is the union of the intervals described above? Explain your answer.
(b) What is the intersection of the intervals described above? Explain your answer.