Binary Relation: Understanding Relationships Between Sets

Overview

A binary relation is a fundamental concept in mathematics and computer science that describes a relationship between elements of two sets, or between elements of a single set and itself. It essentially defines a set of ordered pairs where the first element is related to the second.

Key Concepts

• Definition: A subset of the Cartesian product of two sets (e.g., A x B).
• Domain and Range: The set of first elements and the set of second elements in the ordered pairs, respectively.
• Types: Relations can be reflexive, symmetric, transitive, antisymmetric, etc.

Deep Dive

Consider two sets, A and B. A binary relation R from A to B is a subset of A × B. If (a, b) is in R, we say ‘a is related to b’ by R. If the relation is from a set A to itself, it’s a relation on A.

Properties:

• Reflexive: For all ‘a’ in set A, (a, a) is in R.
• Symmetric: If (a, b) is in R, then (b, a) is also in R.
• Transitive: If (a, b) is in R and (b, c) is in R, then (a, c) is also in R.

Applications

Binary relations are used extensively:

• Mathematics: Defining orderings (like ≤), equivalence relations, and functions.
• Computer Science: Database schemas, graph theory (representing connections), and formal logic.
• Logic: Modeling logical connectives and inference rules.

Challenges & Misconceptions

A common misconception is confusing a relation with a function. While all functions are binary relations, not all binary relations are functions (a function requires each input to have exactly one output).

FAQs

What is an example of a binary relation? The relation ‘is less than’ (<) between two sets of numbers is a classic example. For sets A={1, 2} and B={3, 4}, the relation R = {(1, 3), (1, 4), (2, 3), (2, 4)}.

How do we represent binary relations? They can be represented using ordered pairs, matrices, or directed graphs.