Introduction to Sets
This section introduces the fundamental concept of a set in mathematics. A set is a well-defined collection of distinct objects, known as elements. Explore how sets are defined and represented.
Definition and Representation
A set is defined by its elements, where order does not matter and duplicates are ignored. For example, `{1, 2, 3}` is the same as `{3, 1, 2}`. Sets can be represented by listing elements (Roster Notation) or by describing a property they share (Set-Builder Notation).
Roster Notation
e.g., `{1, 2, 3, 4}`
Explicitly lists all elements.
Set-Builder Notation
e.g., `{x | x is a positive integer and x < 5}`
Describes a rule for membership.
Interactive Set Operations
Explore fundamental set operations by defining two sets below. The tool will calculate the result of various operations and visualize the relationship between the sets. This provides a hands-on way to understand how sets can be combined and manipulated.
Result:
{}
Set Visualization
Relation Explorer
A relation describes a connection between elements of sets. This tool allows you to define a set and a binary relation on it. It will then automatically generate the matrix and directed graph (digraph) representations and analyze its fundamental properties.
Relation Properties
Zero-One Matrix
Directed Graph (Digraph)
Equivalence Relations & Classes
An equivalence relation partitions a set into disjoint subsets called equivalence classes. Elements within the same class are considered "equivalent." This tool demonstrates this concept by identifying the classes for a given equivalence relation.
An equivalence relation must be Reflexive, Symmetric, and Transitive. Let's explore the classic example: congruence modulo n.
Equivalence Classes:
Partial Orders & Hasse Diagrams
A partial order establishes a hierarchy on a set. Unlike a total order, not all elements need to be comparable. A Hasse diagram is a simplified graph for visualizing a partial order. This tool automatically generates one for the "divides" relation.
A partial order must be Reflexive, Antisymmetric, and Transitive. A common example is the "divides" relation on a set of integers.