Binary relation

Definition

A binary relation is a mathematical concept that describes a relationship between two sets, where each element of one set is associated with elements of another set. It forms the foundation for many important concepts in mathematics, including functions, equivalence relations, and order relations. Understanding binary relations allows for a deeper comprehension of how elements interact and relate to each other across different contexts.

5 Must Know Facts For Your Next Test

1. A binary relation on sets A and B can be represented as a subset of the Cartesian product A x B.
2. Binary relations can be represented using matrices or graphs, providing visual insights into their structure.
3. Reflexivity, symmetry, and transitivity are key properties that can characterize specific types of binary relations.
4. Functions are a specific kind of binary relation where every element in the first set is associated with exactly one element in the second set.
5. Binary relations can be finite or infinite, depending on the nature of the sets involved.

Review Questions

• How can you demonstrate whether a given binary relation is reflexive or not?
  • To determine if a binary relation R on a set A is reflexive, you need to check that every element in A is related to itself. This means that for all elements 'a' in A, the pair (a, a) must be included in the relation R. If there exists even one element for which (a, a) is not present in R, then the relation is not reflexive.
• What distinguishes an equivalence relation from a general binary relation?
  • An equivalence relation is a specific type of binary relation that must satisfy three properties: reflexivity (every element relates to itself), symmetry (if one element relates to another, then the second relates back), and transitivity (if one element relates to a second and that second relates to a third, then the first must relate to the third). These properties allow equivalence relations to partition sets into disjoint equivalence classes, which is not guaranteed in general binary relations.
• Evaluate the significance of binary relations in understanding functions and their properties.
  • Binary relations are crucial in defining functions since every function can be viewed as a binary relation where each input from the domain corresponds to exactly one output in the codomain. This relationship helps establish important properties such as injectivity (one-to-one), surjectivity (onto), and bijectivity (both one-to-one and onto). Understanding how these functions behave as binary relations allows mathematicians to explore deeper concepts like continuity and limits in more advanced mathematical analysis.

Related terms

Cartesian Product:

The Cartesian product is the set of all ordered pairs formed by taking one element from each of two sets.

Equivalence Relation:

An equivalence relation is a special type of binary relation that satisfies reflexivity, symmetry, and transitivity, grouping elements into equivalence classes.

Partial Order:

A partial order is a binary relation that is reflexive, antisymmetric, and transitive, allowing for some elements to be comparable while others are not.