Set Theory Fundamentals: Definitions, Notation and Examples

1.1 Theory of Sets

Theory of sets. Why is it said to be a mathematical system and language? It is said to be a mathematical system because it contains a set of operations, theorems, functions and relations, and it underpins areas such as algebra, geometry, calculus and more.

Set theory is an appropriate tool for structured thinking and for developing the capacity to analyze and design solutions for particular problems. It allows focusing on a problem as a whole by removing what is irrelevant and highlighting the essentials.

Set theory facilitates the visualization of relationships between all component parts of a problem as well as each part individually. It lets us combine elements within its own methodology and use deductive reasoning together with analytical ability to manage information.

Main approach to problem solving

What is the main approach in solving problems? To analyze the available information, evaluate and separate the important from the irrelevant, apply a consistent methodology, and find a solution to support decision making.

1.2 How to visualize relationships

How is it easier to visualize relationships? By using closed figures such as rectangles, squares, circles, Venn diagrams, triangles, etc., we can represent sets and their intersections, unions and differences more clearly.

History and origin of Boolean algebra

Who created it and when? It was created by the mathematician George Boole; hence the system is called Boolean algebra (also called Boolean logic). Boolean algebra was developed in the mid-19th century, with key works published by Boole in 1847 and 1854.

1.3 Definition of a Set and Requirements

Definition: A set is a collection of objects: people, things, animals, numbers, etc., with or without any relationship between them.

The essential requirements to form a set are:

1. Well-defined membership: The collection must be well defined so that for any object it is clear whether it belongs to the set or not. Example: “all taxi drivers” is ambiguous; a better description is “all taxi drivers in the city of Matamoros.”
2. Distinct elements: Elements of the set must be considered without repetition. If an element appears multiple times, it is counted once. Example: the distinct letters in the word "Mississippi" give M = {m, i, s, p}; the number of distinct elements n = 4.

1.4 Role of order in sets

What role does order play in sets? The order in which elements of a set are listed is irrelevant. Example: M = {m, i, s, p}, N = {i, m, s, p}, O = {s, p, i, m} -- all represent the same set.

Types of set notation

Notation types of sets:

1. Roster (list) method: List the elements explicitly. Example: A = {o, e, a, i, u} (note: duplicates are not repeated in set notation).
2. Set-builder (description) method: Describe properties that define the elements. Example: A = { x | x is a vowel of the alphabet }.

Adopted symbology

Adopted symbology: In set theory, sets are typically denoted by uppercase letters (A, B, Z) and their elements by lowercase letters (a, b, x, y). For example:

• B = {b0, b1, b2, ...}
• Example of element notation: h, e, b

Use standard symbols such as { } for sets, ∈ for membership, ∉ for non-membership, and set operators like ∪ (union), ∩ (intersection), and \ (set difference) when describing relationships.