1. Introduction to Set Theory

Introduction to Set Theory

Basics of Sets, Relations, and Functions

Set: A set is a collection of distinct objects, called elements. For example, \( A = \{1, 2, 3\} \) represents a set with elements 1, 2, and 3.

Relations: A relation between sets A and B is a subset of their Cartesian product \( A \times B \). Common types of relations include:

• Binary relation: A relation between two sets.
• Equivalence relation: A relation that is reflexive, symmetric, and transitive.
• Order relation: A relation that is reflexive, antisymmetric, and transitive.

Functions: A function is a relation between two sets such that each element of the first set (domain) is associated with exactly one element of the second set (codomain). Notation: \( f: A \rightarrow B \), where \( A \) is the domain and \( B \) is the codomain.

Operations on Sets

Unions, Intersections, and Complements:

• Union (\( A \cup B \)): The union of sets A and B contains all elements that are in A, in B, or in both.
• Intersection (\( A \cap B \)): The intersection of sets A and B contains all elements that are common to both A and B.
• Complement (\( A^c \)): The complement of set A contains all elements that are not in A, within some universal set.

Cartesian Products and Power Sets

Cartesian Product: The Cartesian product of sets A and B, denoted by \( A \times B \), is the set of all ordered pairs \( (a, b) \) where \( a \) is in A and \( b \) is in B.

Power Set: The power set of a set A, denoted by \( \mathcal{P}(A) \) or \( 2^A \), is the set of all subsets of A, including the empty set and A itself.