CLOSURES OF RELATIONS

Lesson Title: Determining Reflexive, Symmetric, and Transitive Closures of Relations

Introduction: Relations are fundamental in the study of mathematics and computer science. Understanding their properties, such as reflexivity, symmetry, and transitivity, is essential in various applications, including graph theory, database design, and formal logic. In this lesson, we will explore how to determine the reflexive, symmetric, and transitive closures of relations.

Section 1: Reflexive Closure

What is a Reflexive Relation?

A relation R on a set A is reflexive if for every element a in A, (a, a) is in R. In other words, every element of the set relates to itself.

Determining the Reflexive Closure:

To find the reflexive closure of a relation R, follow these steps:

• Start with the original relation R.

• For each element a in the set A, check if (a, a) is already in R. If it's not, add (a, a) to R.

• The resulting relation is the reflexive closure of R.

Example:

Let's say we have a relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}. To find the reflexive closure:

• Add (1, 1), (2, 2), and (3, 3) to R, making it {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}.

Section 2: Symmetric Closure

What is a Symmetric Relation?

A relation R on a set A is symmetric if for every (a, b) in R, (b, a) is also in R. In other words, if (a, b) is related, then (b, a) must also be related.

Determining the Symmetric Closure:

To find the symmetric closure of a relation R, follow these steps:

• Start with the original relation R.

• For each pair (a, b) in R where a ≠ b, add (b, a) to R if it's not already there.

• The resulting relation is the symmetric closure of R.

Example:

Using the relation R from the previous example {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}, the symmetric closure is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}.

Section 3: Transitive Closure

What is a Transitive Relation?

A relation R on a set A is transitive if for every (a, b) and (b, c) in R, (a, c) is also in R. In other words, if there is a chain of relations from a to b and from b to c, then there must also be a relation from a to c.

Determining the Transitive Closure:

To find the transitive closure of a relation R, follow these steps:

• Start with the original relation R.

• For each pair (a, b) and (b, c) in R, add (a, c) to R if it's not already there.

• Repeat step 2 until no more elements can be added.

• The resulting relation is the transitive closure of R.

Example:

Using the relation R from the symmetric closure example {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}, the transitive closure is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}.

Conclusion:

In this lesson, we've explored how to determine the reflexive, symmetric, and transitive closures of relations. Understanding these closure properties is vital in various mathematical and computational contexts, including graph theory, formal logic, and database design. Practicing these concepts will enhance your ability to analyze and manipulate relations effectively.