elementary-set-theory

Elementary Set Theory is the mathematical foundation for defining, organizing, and operating on collections of objects.

It was pioneered by [[Georg Cantor]] and is fundamental to almost every area of modern mathematics

Definition of a Set

A set is a well-defined collection of distinct objects.

Objects/Elements: The objects belonging to a set are called its elements or members.

Notation: Sets are typically denoted by capital letters (e.g., A, B, S) and their elements are enclosed in curly braces { }.

Roster Method: Listing all elements: A = {1, 2, 3, 4}.

Set-Builder Notation: Describing a property: $$B = {x \mid x \text{ is an even integer}} $$This is read as “the set of all x such that x is an even integer.”

Membership: $x \in A$ means “x is an element of set A.” $y \notin A$ means “y is not an element of set A.”

Types of Sets

Finite Set

• A set whose elements can be counted, resulting in a specific natural number.
• $A = {a,b,c}$

Infinite Set

• A set whose elements cannot be counted (e.g., the set of all integers).
• $\mathbb{Z} = {\dots, -1, 0, 1, \dots}$

Empty Set $(\emptyset or {})$

• A unique set containing no elements.
• $C = {x \mid x \in \mathbb{R} \text{ and } x^2 = -1}$

Universal Set $(U)$

• The set containing all objects or elements relevant to a particular context or problem.
• For number theory, U might be $\mathbb{Z}$ (Integers)

Cardinality

The cardinality of a finite set A, denoted by $|A|$ or $n(A)$, is the number of distinct elements in the set.

Example: If $S = {a, e, i, o, u}$, then $|S| = 5$