Mathematical Foundations for Computing

2.

Basic Structures: Sets, Functions, and Relations

2.1.

2.1.1.

2.1.1.1.

2.1.1.2.

Set-Builder Notation

2.1.1.3.

Describing Infinite Sets

2.1.1.4.

Membership and Non-membership

2.1.2.

2.1.2.1.

2.1.2.2.

2.1.2.3.

2.1.2.4.

2.1.2.5.

Natural Numbers

2.1.2.6.

2.1.2.7.

Rational Numbers

2.1.2.8.

2.1.3.

Subsets and Proper Subsets

2.1.3.1.

Definition of Subset

2.1.3.2.

2.1.3.3.

Subset Notation

2.1.3.4.

Subset Relationships

2.1.4.

2.1.4.1.

2.1.4.2.

2.1.4.3.

2.1.4.4.

2.1.4.5.

Symmetric Difference

2.1.4.6.

2.1.5.

2.1.5.1.

Representing Set Operations

2.1.5.2.

Visualizing Relationships

2.1.5.3.

Three-Set Venn Diagrams

2.1.6.

Set Cardinality

2.1.6.1.

Counting Elements in Finite Sets

2.1.6.2.

2.1.6.3.

2.1.6.4.

Uncountable Sets

2.1.6.5.

Comparing Cardinalities

2.1.6.6.

Cantor's Theorem

2.1.7.

Cartesian Products

2.1.7.1.

Definition and Notation

2.1.7.2.

Properties of Cartesian Products

2.1.7.3.

n-ary Cartesian Products

2.1.7.4.

Ordered Pairs and Tuples

2.1.8.

2.1.8.1.

Proving Set Identities

2.1.8.2.

Common Set Laws

2.1.8.3.

Using Set Identities in Proofs

2.2.

2.2.1.

Definition of a Function

2.2.1.1.

Mapping from Domain to Codomain

2.2.1.2.

Function Notation

2.2.1.3.

Function Equality

2.2.2.

Domain, Codomain, and Range

2.2.2.1.

Identifying Domain and Codomain

2.2.2.2.

Determining the Range

2.2.2.3.

Image and Preimage

2.2.3.

Types of Functions

2.2.3.1.

Injective Functions

2.2.3.2.

Surjective Functions

2.2.3.3.

Bijective Functions

2.2.3.4.

Constant Functions

2.2.3.5.

Identity Function

2.2.3.6.

Partial Functions

2.2.4.

Inverse Functions

2.2.4.1.

Definition and Existence

2.2.4.2.

Finding Inverses

2.2.4.3.

Properties of Inverse Functions

2.2.4.4.

Left and Right Inverses

2.2.5.

Composition of Functions

2.2.5.1.

Definition of Function Composition

2.2.5.2.

Associativity of Composition

2.2.5.3.

Domain and Range of Composed Functions

2.2.5.4.

Composition with Inverse Functions

2.2.6.

Important Functions in Computer Science

2.2.6.1.

2.2.6.2.

Ceiling Function

2.2.6.3.

Factorial Function

2.2.6.4.

Logarithmic Functions

2.2.6.5.

Exponential Functions

2.2.6.6.

Polynomial Functions

2.2.6.7.

Boolean Functions

2.3.

2.3.1.

Binary Relations

2.3.1.1.

Definition and Examples

2.3.1.2.

n-ary Relations

2.3.1.3.

Relation as a Set of Ordered Pairs

2.3.2.

Representing Relations

2.3.2.1.

2.3.2.2.

Using Directed Graphs

2.3.2.3.

Relation Tables

2.3.2.4.

2.3.3.

Properties of Relations

2.3.3.1.

2.3.3.2.

2.3.3.3.

2.3.3.4.

2.3.3.5.

2.3.3.6.

2.3.3.7.

Combining Properties

2.3.4.

Combining Relations

2.3.4.1.

Union of Relations

2.3.4.2.

Intersection of Relations

2.3.4.3.

Composition of Relations

2.3.4.4.

Inverse of a Relation

2.3.4.5.

Powers of Relations

2.3.5.

Equivalence Relations

2.3.5.1.

Definition of Equivalence Relation

2.3.5.2.

Equivalence Classes

2.3.5.3.

Partitions Induced by Equivalence Relations

2.3.5.4.

2.3.6.

Partial Orderings

2.3.6.1.

Definition of Partial Order

2.3.6.2.

Partially Ordered Sets

2.3.6.3.

2.3.6.4.

Maximal and Minimal Elements

2.3.6.5.

Greatest and Least Elements

2.3.6.6.

Upper and Lower Bounds

2.3.6.7.

Total Orderings

2.3.6.8.

Well-Ordered Sets

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