Relations - Examples

Based on the sources and our conversation history, a relation is defined as a collection of ordered pairs or n-tuples from a Cartesian product of sets. It is formally a subset of this Cartesian product. Relations are used to describe connections or relationships between elements.

Here are several examples of relations discussed in the sources:

• Relations on Numbers:

  • A relation between two sets of natural numbers, N, could be the set of pairs (n, m) where m is 1 plus n. This results in pairs like (0, 1), (2, 3), (17, 18), etc., which form a subset of N × N.
  • The divisibility relation on natural numbers (N × N) or integers (Z × Z) is a relation where a pair (d, n) is included if d is a factor of n (meaning n mod d = 0). For example, (2, 82) and (14, 56) would be in this relation. This relation can be defined using set comprehension as the set of pairs (d, n) in N × N such that d divides n.
  • A relation could be defined by pairs of integers (p, q) where their greatest common divisor is 1. This is related to the concept of reduced fractions.
  • The “less than” relation (<) on numbers is a relation where a pair (a, b) is included if a is less than b. Similarly, the “strictly less than” relation is also an example.
  • An equivalence relation example is connecting numbers that have the same remainder modulo 5. Numbers like 7 and 22 are related in this way because both have a remainder of 2 when divided by 5.
  • A relation on the Cartesian product N × R could be the set of pairs (m, r) such that r is the square root of m. Pairs like (2, √2), (4, 2), (7, √7) would be in this relation.
  • The relation associated with a function f(x) = x² is the set of pairs (x, y) such that y = x². When plotted, these points form the graph of the function, a parabola.

• Relations between Different Sets:

  • An allocation relation between a set of teachers (T) and a set of courses (C) in a school could be the set of pairs (t, c) where teacher ’t’ is actually teaching course ‘c’. This relation would be a subset of the Cartesian product T × C.

• Relations on the Same Set (Binary Relations):

  • The identity relation on a set A is the set of all pairs (a, a) where ‘a’ is an element of A. This represents the property of equality. For natural numbers (N × N), this would be pairs like (0, 0), (1, 1), etc..
  • The “mother of” relation between people in a set P would be the set of pairs (m, c) such that ’m’ is the mother of ‘c’. This is a subset of P × P. This relation is later noted as being a function because every person has exactly one mother.
  • A relation on the set of real numbers (R × R) could be the set of points (a, b) that are at a distance of 5 from the origin (0, 0). This satisfies the equation a² + b² = 5² and defines a circle of radius 5 centred at the origin when plotted.

• Relations with More Than Two Sets:

  • A relation could involve more than just pairs (binary relations). For example, a relation could describe four points in R² that form the corners of a square. This would be a relation on (R²)⁴, meaning it involves 4-tuples where each element of the tuple is itself a pair of real numbers (an x and y coordinate). In general, relations can involve n-tuples from the Cartesian product of n sets.

• Relations in Computing and Data Science:

  • Information about an airline’s routes can be represented as a relation. A set D of direct flights between cities in a set C is a relation, a subset of C × C.
  • This airline example can be extended to include distance, becoming a relation on C × C × Natural Numbers, where a tuple (city1, city2, distance) is included if there’s a direct flight between city1 and city2 with that distance.
  • Crucially, tables in computing and data science are effectively relations.
    • A student record table with columns for roll number, name, and date of birth can be seen as a relation.
    • A grades table with columns for roll number, subject, and grade is also a relation.
  • Operations like the Join operation, which combines data from different tables, are described as operations on relations.

These examples illustrate the broad applicability of the concept of a relation, from abstract mathematical definitions to concrete representations in data systems.