Relations
Based on the provided sources and our conversation history, a relation is fundamentally a way to describe connections or relationships between elements from one or more sets.
Here’s a breakdown of what the sources say about relations:
- A relation is a collection of ordered pairs, or more generally, n-tuples of elements drawn from a Cartesian product of sets.
- They allow us to construct new sets from old sets.
- A relation is formally defined as a subset of a Cartesian product of sets. For example, a relation between two sets X and Y is a subset of X × Y.
- The order of elements is important in the ordered pairs (or n-tuples) that form a relation. For instance, the pair (a, b) is distinct from (b, a) unless a = b.
- Relations can be defined either by explicitly listing the pairs (for finite relations) or implicitly using set comprehension notation, which applies a filter or condition to elements from the Cartesian product.
- A relation describes special tuples within a Cartesian product that satisfy a certain condition or are of interest to us.
Visualising Relations:
- For relations on numbers (like natural numbers or real numbers), they can be visualised by plotting the pairs as points on a grid or graph.
- Relations can also be represented using graphs with nodes representing elements and arrows connecting related pairs.
- Importantly, tables are also a way of representing relations, particularly relevant in computing and data science.
Properties of Binary Relations:
Binary relations (relations defined on two copies of the same set, R ⊆ A × A) can possess certain properties:
- Reflexive relation: For every element ‘a’ in the set, the pair (a, a) must be in the relation R. This means every element is related to itself. The identity relation (the set of all pairs (a, a)) is included in any reflexive relation.
- Irreflexive relation: For every element ‘a’ in the set, the pair (a, a) is never in the relation R.
- Symmetric relation: If the pair (x, y) is in the relation R, then the pair (y, x) must also be in R. An example is the distance between two points, which is the same regardless of the order. Symmetric relations can be represented more efficiently in tables by recording only half the entries.
- Anti-symmetric relation: If the pair (a, b) is in R (where a and b are distinct elements), then the pair (b, a) should not be in R. The “strictly less than” (<) relation is an example of an anti-symmetric relation.
- Transitive relation: If (x, y) is in R and (y, z) is in R, then (x, z) must also be in R. The divisibility relation and the “less than” relation are examples. Transitive closure is a concept related to transitivity, representing all possible reachability relationships (paths) between vertices in a directed graph.
Equivalence Relations:
- A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
- Equivalence relations partition a set into disjoint equivalence classes. Elements within the same class are related to each other, while elements in different classes are not. They represent a type of equality.
Relation to Functions:
- Functions are closely related to relations.
- A function is a special type of relation where each element in the domain set (input) is paired with exactly one element in the codomain/range set (output).
- Every function ‘f’ has an associated relation, which consists of all the input-output pairs (x, f(x)). Plotting the graph of a function is essentially plotting the points of this associated relation.
Relations in Computing and Data Science:
- Relations are highly relevant in computing and data science because tables (like those in databases) are essentially relations.
- Operations performed on tables, such as the Join operation, are operations on relations.
- Understanding relations is important for working with and combining data stored in tabular form.
Relations in Graph Theory:
- Relations are used to model pairwise relationships between objects, a core concept in graph theory.
- In a directed graph, edges are ordered pairs representing a relation.
- Concepts like reachability and transitive closure in graphs are based on the transitivity property of relations.
- Matrices, such as the adjacency matrix, can be used to represent relations in graphs. Matrix multiplication is a tool for analyzing these relations, for example, to find paths and compute transitive closure.
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