Sets

A set is defined as a collection of distinct objects grouped together by a clear and objective criterion that unequivocally determines whether any given object belongs to that set. These objects are known as elements.

In mathematics, a set is usually denoted by an uppercase letter from either the Greek or Latin alphabet (e.g., A, B, C, ...), while its elements are represented by lowercase letters (e.g., a, b, c, ...) between two curly braces.

$$ A = \{ 1, 2, 3, 4, 5, ... \} $$

Each element in a set is unique and, unlike in vectors or ordered sequences, is not arranged in any particular order.

The symbol "∈" indicates membership, showing that an object or element is part of a set. Conversely, "∉" is used to denote that an object does not belong to a set.

Example: Consider a set A that represents all positive integers less than 5: $$ A = \{1, 2, 3, 4\} $$. Here, set A contains four elements. Taking elements p=3 and q=5, we can state: $$ p ∈ A $$ $$ q ∉ A $$. This explicitly shows that the number 3 is included in set A, whereas number 5 is not. If we also consider another element, say r=2, we can succinctly indicate that both p and r belong to set A: $$ p, r ∈ A $$.

Set Membership Criteria

A set is characterized by a membership criterion that clearly defines whether an object belongs to or does not belong to a set. This is symbolized by ∈ for membership. For example, the membership of an element x in set Y is notated as "x ∈ Y," meaning "x belongs to Y".

$$ x \in Y $$

Similarly, non-membership is denoted by the barred symbol ∉. Non-membership of an element x in set Z is written as "x ∉ Z" and interpreted as "x does not belong to Z".

$$ x \not \in Z $$

Different notations and representations are employed to define a mathematical set comprehensively.

Representing a Set

Graphical representation often utilizes Euler-Venn diagrams, which show how sets overlap and intersect through circles or other shapes. These diagrams effectively illustrate the relationships between sets in an intuitive manner.

Other forms of representation include tabular, enumerative, and characteristic property-based methods.

• Formal notation
  We employ a membership criterion to specify set elements. For instance, \( x \in \mathbb{N} \) suggests \( x \) is a natural number, and \( x < 100 \) limits \( x \) to values less than 100. $$ A = \{ x \in \mathbb{N} : x < 100 \} $$ This set encompasses all natural numbers from 0 to 99.
• Enumerative notation
  This approach directly lists the set's elements, especially useful when the set is small and elements can be easily enumerated. For instance, a set containing natural numbers less than 100 might be defined as: $$ A = \{0, 1, 2, 3, ..., 98, 99\} $$. This list starts at 0 and includes every number up to 99, because 100 does not meet the condition x<100.

Both notational methods are effective and chosen based on the context and the level of clarity required. Formal notation is favored in more rigorous mathematical settings and for larger sets, while enumerative notation is more accessible and visually straightforward for examples and basic concept instruction.

Numerical Sets

In mathematics, specific symbols are used to represent standard numerical sets, ensuring immediate clarity on the types of numbers discussed. Below are the symbols for major numerical sets:

• \( \mathbb{N} \) represents all natural numbers, which are non-negative integers starting from 0 or 1, depending on the convention used (0, 1, 2, 3, ...). In some contexts, \( \mathbb{N} \) begins at 1, excluding 0.
• \( \mathbb{Z} \) denotes all integers, encompassing both positive and negative numbers, including zero (..., -3, -2, -1, 0, 1, 2, 3, ...).
• \( \mathbb{Q} \) signifies all rational numbers, each expressible as a fraction \( \frac{p}{q} \) where \( q \) is not zero (e.g., \( \frac{1}{2} \), \( \frac{3}{4} \), \( -\frac{5}{3} \), etc.).
• \( \mathbb{R} \) encompasses all real numbers, combining all rational numbers (\( \mathbb{Q} \)) with all irrational numbers (such as \( \sqrt{2} \), \( \pi \), \( e \), etc.), which are not expressible as fractions.
• \( \mathbb{C} \) covers all complex numbers, any of which can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). This set extends the real number field \( \mathbb{R} \) to include solutions to equations without real solutions, like \( x^2 + 1 = 0 \). For instance, in the complex number \( 3 + 4i \), \( 3 \) is the real part and \( 4i \) the imaginary part.

Example: Imagine a diagram with concentric circles to show how these sets intersect and overlap. \( \mathbb{N} \) is the innermost circle, surrounded by \( \mathbb{Z} \), indicating that all natural numbers are integers. \( \mathbb{Q} \) then encircles \( \mathbb{Z} \), as every integer is rational. Lastly, \( \mathbb{R} \) is the outermost circle, containing all others and representing the complete set of real numbers, both rational and irrational. This visualization aids in understanding the relationships among the different numerical sets.

Types of Sets

Sets can be categorized based on the number of their elements.

• Finite set: Consists of a specific number of elements, like the set of months in a year or days in a month.
• Infinite set: Contains an endless number of elements, such as the set of natural numbers or real numbers.
• Empty set: A set with no elements, represented by the symbol Ø. An example is the set of squares with three sides.
• Universal set: Includes all conceivable elements, making every other set a subset of it. It is generally denoted by the uppercase letter U.