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: $f:A\to B$, 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.