Unit 2 - Sets, Functions, and Groups

This lesson provides an overview of Unit 2. You will explore the definitions, properties, and operations of sets, functions, and groups, building a foundation for understanding relations and algebraic structures.

1. Sets

Properties of a Set

What is a Set
Order of a Set

Describing Sets

The Descriptive Method
The Tabular Method
Set-builder Method

Types of Sets

Finite and Infinite Sets
Singleton and Null Sets
Proper and Improper Subsets
Power Set

Operations on Sets

Intersection ( $A \cap B$ )
Union ( $A \cup B$ )
Complement ( $A'$ )
Difference ( $A - B$ )

Properties of Sets

Commutative, Associative, and Distributive Properties
De Morgan's Laws ( $(A \cup B)' = A' \cap B'$ )

2. Logic

Types of Logic

Inductive Logic
Deductive Logic
Symbolic Logic

Symbolic Logic Operations

Negation (~P)
Conjunction (P ∧ Q)
Disjunction (P ∨ Q)
Conditional (P → Q)
Biconditional (P ↔ Q)

Symbolic Logic Terms

Converse
Inverse
Contrapositive
Tautologies
Absurdity

3. Relations and Functions

Relations

Binary Relations
Domain
Range

Functions and Types

Into Function
Onto Function
Injective Function
Bijective Function
One-to-One Function

Operations on Functions

Inverse Functions

Graphing Functions

Linear Functions
Quadratic Functions
Cubic Functions

4. Algebraic Structures

Operations

Unary Operations
Binary Operations
Residue Classes of Modulo ( $R = Q \cup Q'$ )
Properties of Binary Operations

Classification of Sets Based on Binary Operation Properties

Groupoid
Semi-group
Non-Abelian
Monoid
Group

End of Lesson