Sets

Describing a Set

At its core, the concept of a set is one of the most foundational ideas in mathematics, from which nearly all other mathematical structures are built. A set is simply a collection of objects. The objects that make up a set are called its elements or members.

Standard Notation

To discuss sets with precision, mathematicians use a standard symbolic language. The most essential notations are summarized below.

Concept	Symbolic Representation
Designating a set	Capital letters (e.g., $A, B, C$)
Representing an element	Lower-case letters (e.g., $a, b, c$)
An element belongs to a set	$a \in A$ (read as “$a$ is an element of $A$”)
An element does not belong to a set	$a \notin A$ (read as “$a$ is not an element of $A$”)

A critical requirement when describing a set is that the description must make it perfectly clear which elements belong to it. There are several standard methods for achieving this clarity.

Roster Method

For sets with a small number of elements, the most straightforward method is to list them explicitly between braces $\{\}$.

For example, the set $S$ containing the first three positive integers is written as:

$$ S = \{1, 2, 3\} $$

A key insight is that the order in which elements are listed does not change the set.

Ellipsis Notation

When a set contains too many elements to list conveniently, the ellipsis ($\dots$) notation is used. It has two primary applications:

1. “And so on up to”: For a finite set with a clear pattern, the ellipsis indicates continuation up to a final element.

$$ X = \{1, 3, 5, \dots, 49\} $$

represents the set of positive odd integers less than $50$.

2. “And so on” indefinitely: For an infinite set, the ellipsis indicates that the pattern continues without end.

$$ Y = \{2, 4, 6, \dots\} $$

represents the set of all positive even integers.

Set-Builder Notation

A powerful method for defining a set is to describe the property that all its elements must satisfy. This is known as set-builder notation. The general format is:

$$ S = \{x : p(x)\} $$

This is read as “$S$ is the set of all elements $x$ such that the property $p(x)$ is true.”

The Special Case of the Empty Set

A set is not required to contain any elements. The unique set that contains no elements is called the empty set. It is denoted by one of two symbols:

$$ \emptyset $$

or

$$ \{\} $$

Commonly Used Number Sets

Several sets of numbers are so frequently used in mathematics that they are given their own special symbols.

Symbol	Set Name	Description/Example
$\mathbb{N}$	Natural Numbers	The set of positive integers
$\mathbb{Z}$	Integers	The set of positive, negative, and zero integers
$\mathbb{Q}$	Rational Numbers	Numbers that can be expressed as a fraction $\frac{m}{n}$, where $m, n \in \mathbb{Z}$ and $n \neq 0$.
$\mathbb{R}$	Real Numbers	The set of all rational and irrational numbers.
$\mathbb{C}$	Complex Numbers	Numbers of the form $a + bi$, where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$.

Subsets

For example, if $A = \{1, 2, 3\}$, its power set is: $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$

A crucial property of power sets is related to their size, or cardinality. For any finite set $A$, the number of elements in its power set is given by the formula:

$$ |\mathcal{P}(A)| = 2^{|A|} $$

In the example above, $|A| = 3$, so $|\mathcal{P}(A)| = 2^3 = 8$.

Set Operations

Just as numbers can be combined with operations like addition and multiplication, sets can be combined using operations to create new sets. The two most fundamental operations are union and intersection.

• The union of the set of Natural Numbers ($\mathbb{N}$) and the set of Integers ($\mathbb{Z}$) is $\mathbb{Z}$, since every natural number is also an integer. Thus, $\mathbb{N} \cup \mathbb{Z} = \mathbb{Z}$.
• The union of the set of Rational Numbers ($\mathbb{Q}$) and the set of Irrational Numbers ($\mathbb{Q}$) forms the set of all Real Numbers ($\mathbb{R}$). Thus, $\mathbb{Q} \cup \mathbb{I} = \mathbb{R}$.

• The intersection of $\mathbb{N}$ and $\mathbb{Z}$ is $\mathbb{N}$, as the natural numbers are the elements common to both sets. Thus, $\mathbb{N} \cap \mathbb{Z} = \mathbb{N}$.
• The intersection of $\mathbb{Q}$ and $\mathbb{R}$ is $\mathbb{Q}$, since every rational number is also a real number. Thus, $\mathbb{Q} \cap \mathbb{R} = \mathbb{Q}$.

When two sets have no elements in common, their intersection is the empty set ($\emptyset$). Such sets are defined as being disjoint.

The difference $\mathbb{R} − \mathbb{Q}$ results in the set of Irrational Numbers ($\mathbb{I}$), as it includes all real numbers that are not rational.

If the universal set is the set of all Integers ($U = \mathbb{Z}$), then the complement of the Natural Numbers ($\mathbb{N}$) is $\overline{\mathbb{N}} = \{0, −1, −2, \dots\}$.

If the universal set is the set of all Real Numbers ($U = \mathbb{R}$), then the complement of the Rational Numbers ($\mathbb{Q}$) is the set of Irrational Numbers ($\mathbb{I}$). Thus, $\overline{\mathbb{Q}} = \mathbb{I}$.

Indexed Collections of Sets

While we can manually write $A \cup B \cup C$, this approach is untenable for dozens or an infinite number of sets. To manage such scenarios we require a more scalable and rigorous notation. This is accomplished through the use of indexed collections, which provide a formal mechanism for addressing any number of sets simultaneously.

The symbol $\alpha$ serves as a dummy variable, allowing us to reference any specific set within the collection.

For example, for each natural number $n \in \mathbb{N}$, define the set $A_n$ as the closed interval $[-\frac{1}{n}, \frac{1}{n}]$.

The union results in $[−1, 1]$ because the collection of sets is nested ($\dots \subset A_3 \subset A_2 \subset A_1$). In such a nested sequence, the union is simply the largest set in the collection, which is $A_1 = [−1, 1]$.

Therefore:

$$ \bigcup_{n \in \mathbb{N}} A_n = [−1, 1] $$

For example, re-examining the collection $A_n = [−\frac{1}{n}, \frac{1}{n}]$, the result

$$ \bigcap_{n \in \mathbb{N}} A_n = \{0\} $$

is significant. To understand it, consider any non-zero real number, $x$. No matter how small $x$ is, we can always find a natural number $n$ large enough such that $\frac{1}{n} < |x|$.

For this $n$, $x$ is not in the interval $A_n$, and thus it cannot be in the intersection. The only number that remains in every interval as $n$ approaches infinity is $0$.

Partitions of Sets

Partitioning is a formal method for dividing a whole into a collection of non-overlapping, exhaustive parts. This concept is fundamental to classification problems, data clustering, and parallel processing.

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. The collection $S = \{\{1, 6\}, \{2, 5\}, \{4, 7\}\}$ is pairwise disjoint because the intersection of any two distinct sets from $S$ is $\emptyset$ (e.g., $\{1, 6\} \cap \{2, 5\} = \emptyset$).

The ’non-empty’ and ‘complete union’ properties together ensure that every element of $A$ belongs to at least one subset. The ‘pairwise disjoint’ property ensures that no element can belong to more than one. Combined, these three rigorous checks guarantee that every element of $A$ belongs to exactly one subset.

The concept of partitions extends to well-known infinite sets, providing fundamental classifications:

The set of integers $\mathbb{Z}$ can be partitioned into the set of even integers and the set of odd integers.

The set of real numbers $\mathbb{R}$ can be partitioned into the set of positive real numbers ($\mathbb{R}^+$), the set of negative real numbers, and the singleton set $\{0\}$.

The set of real numbers $\mathbb{R}$ can also be partitioned into the set of rational numbers ($\mathbb{Q}$) and the set of irrational numbers ($\mathbb{I}$).

Cartesian Products of Sets

The Cartesian product is a fundamental set operation that creates a new set from two or more existing sets by forming ordered pairs of their elements.

A critical distinction must be made between a set and an ordered pair. For a set containing two elements, such as $\{x, y\}$, the order is irrelevant, meaning $\{x, y\}$ is identical to $\{y, x\}$. In contrast, for an ordered pair $(x, y)$, the order is fundamental.

In the ordered pair $(x, y)$, $x$ is designated as the first coordinate and $y$ as the second coordinate.

The formal condition for the equality of two ordered pairs is that their corresponding coordinates must be identical. Consequently, if $x \neq y$, then the ordered pair $(x, y)$ is distinct from the ordered pair $(y, x)$.

The Cartesian product is not a commutative operation.

A primary example of the Cartesian product is its application in geometry. The set $\mathbb{R} \times \mathbb{R}$ (the Cartesian product of the set of real numbers with itself) represents the set of all points in the two-dimensional Euclidean plane. This structure allows for the algebraic description of geometric figures.