Write The Relation As A Set Of Ordered Pairs

Write the Relation as a Set of Ordered Pairs: A Comprehensive Guide

Representing relations as sets of ordered pairs is a fundamental concept in mathematics, particularly in set theory and discrete mathematics. Understanding this representation allows for a precise and unambiguous way to define and manipulate relations, laying the groundwork for more advanced topics like functions, graphs, and databases. This comprehensive guide will explore this concept in detail, covering various aspects and providing illustrative examples to solidify your understanding.

What is a Relation?

Before delving into the representation using ordered pairs, let's define what a relation actually is. In simple terms, a relation is a connection or correspondence between elements of two (or more) sets. It describes how elements from one set are related to elements in another set (or even the same set). This relationship can be anything – equality, inequality, divisibility, membership, or any other defined connection.

For example, consider the set of integers A = {1, 2, 3} and the set of even integers B = {2, 4, 6}. We can define a relation 'is less than' ( < ) between A and B. This relation would connect elements from A to elements in B where the element in A is less than the element in B.

Another example: consider the set of students in a class and the set of their assigned grades. The relation could be "student received grade." Each student is related to one or more grades they received.

Representing Relations as Sets of Ordered Pairs

The most common and rigorous way to represent a relation is as a set of ordered pairs. An ordered pair is denoted as (a, b), where 'a' is an element from the first set (often called the domain) and 'b' is an element from the second set (often called the codomain or range). The order of the elements within the pair is crucial; (a, b) is different from (b, a) unless a = b.

The relation is then represented by the set of all such ordered pairs that satisfy the defined relationship. Let's illustrate this with examples:

Example 1: The 'less than' relation

Let's revisit our example of the sets A = {1, 2, 3} and B = {2, 4, 6}, with the relation "is less than" (<). The relation as a set of ordered pairs would be:

R = {(1, 2), (1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6)}

Notice that each pair (a, b) in R satisfies the condition a < b, where a ∈ A and b ∈ B.

Example 2: A relation on a single set

Relations can also be defined on a single set. Consider the set C = {1, 2, 3, 4, 5}. Let's define the relation 'is divisible by' (|) on C. The relation as a set of ordered pairs would be:

S = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (2, 4), (1, 5)}

Here, each pair (a, b) in S satisfies the condition a | b (a divides b), where a, b ∈ C. Note that (1,1), (2,2), etc are included because any number is divisible by itself.

Example 3: A relation representing a function

Functions are a special type of relation where each element in the domain is related to exactly one element in the codomain. Let's consider the function f(x) = x² with the domain D = {1, 2, 3} and the codomain (range) being the set of non-negative integers. The relation representing this function as a set of ordered pairs is:

F = {(1, 1), (2, 4), (3, 9)}

Each element in D is associated with exactly one element in the range.

Properties of Relations

Relations can possess various properties that characterize their nature. These properties are crucial in classifying and understanding different types of relations. The most common properties are:

• Reflexive: A relation R on a set A is reflexive if for every element a ∈ A, (a, a) ∈ R. In simpler terms, every element is related to itself. (e.g., the "is equal to" relation is reflexive).

• Symmetric: A relation R on a set A is symmetric if for every pair (a, b) ∈ R, (b, a) ∈ R. If a is related to b, then b is related to a. (e.g., the "is equal to" relation and the "is the same age as" relation are symmetric).

• Transitive: A relation R on a set A is transitive if for every three elements a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. If a is related to b, and b is related to c, then a is related to c. (e.g., the "less than" relation and the "is an ancestor of" relation are transitive).

• Anti-symmetric: A relation R on a set A is anti-symmetric if for every pair (a, b) ∈ R, and (b, a) ∈ R, then a = b. If a is related to b and b is related to a, then a and b must be the same element. (e.g., the "less than or equal to" relation is anti-symmetric).

Let's illustrate these properties with an example:

Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}.

• Reflexive: Yes, because (1,1), (2,2), and (3,3) are all in R.
• Symmetric: Yes, because if (1,2) ∈ R, then (2,1) ∈ R.
• Transitive: Yes, because we don't have any instances where (a,b) and (b,c) are in R, but (a,c) is not.
• Anti-symmetric: No, because (1,2) and (2,1) are in R, but 1 ≠ 2.

Representing Relations using Matrices and Digraphs

While sets of ordered pairs provide a precise mathematical representation, relations can also be visualized using matrices and digraphs (directed graphs).

Adjacency Matrices:

An adjacency matrix is a square matrix where the rows and columns represent the elements of the set. A '1' in the (i, j) position indicates that element i is related to element j, and a '0' indicates that they are not related.

For Example 2 (relation S), the adjacency matrix would be:

   1 2 3 4 5
1  1 0 0 0 1
2  0 1 0 1 0
3  0 0 1 0 0
4  0 0 0 1 0
5  0 0 0 0 1

Digraphs:

A digraph (directed graph) represents a relation using nodes (vertices) to represent the elements of the set and directed edges (arrows) to represent the relation. An arrow from node a to node b indicates that (a, b) is in the relation.

For Example 2 (relation S), the digraph would show nodes 1, 2, 3, 4, 5, with directed edges from 1 to 1, 2 to 2, 3 to 3, 4 to 4, 5 to 5, 2 to 4, and 1 to 5.

Applications of Relations

The concept of relations as sets of ordered pairs has wide-ranging applications in various fields:

• Databases: Relational databases are fundamentally based on relations. Tables in a database represent relations, with each row representing an ordered pair of attribute values.

• Graph Theory: Graphs and networks are inherently based on relations. The edges in a graph represent the relation between the nodes.

• Computer Science: Relations are used extensively in programming, particularly in data structures and algorithms involving sets and mappings. For example, functions and dictionaries are implementations of relations.

• Formal Language Theory: In computer science, relations are a key component of formal language theory which can describe and manage the structure of programming languages and data formats.

Conclusion

Understanding the representation of relations as sets of ordered pairs is a cornerstone of discrete mathematics and has far-reaching implications across various fields. Mastering this concept, along with the associated properties and visualization techniques, is essential for anyone working with data, algorithms, and mathematical structures. This comprehensive guide has provided a detailed explanation, practical examples, and applications to enhance your comprehension and proficiency in this fundamental mathematical idea. Further exploration into specific types of relations (e.g., equivalence relations, partial orderings) will build upon this foundation, revealing even more profound connections within the mathematical landscape.