*66*

# Closure Properties of Relations

Consider a given set A, and the collection of all relations on A. Let P be a property of such relations, such as being symmetric or being transitive. A relation with property P will be called a P-relation. The P-closure of an arbitrary relation R on A, indicated P (R), is a P-relation such that

**(1) Reflexive and Symmetric Closures:** The next theorem tells us how to obtain the reflexive and symmetric closures of a relation easily.

**Theorem:** Let R be a relation on a set A. Then:

- R âˆª âˆ†
_{A}is the reflexive closure of R - R âˆª R
^{-1}is the symmetric closure of R.

**Example1:**

Find the reflexive closure of R.

**Solution:** R âˆª âˆ† is the smallest relation having reflexive property, Hence,

R_{F}= R âˆª âˆ† = {(k, k), (k, l), (l, l), (l, m), (m, m), (m, k)}.

**Example2:** Consider the relation R on A = {4, 5, 6, 7} defined by

Find the symmetric closure of R.

**Solution:** The smallest relation containing R having the symmetric property is R âˆª R^{-1},i.e.

R_{S}= R âˆª R^{-1}= {(4, 5), (5, 4), (5, 5), (5, 6), (6, 5), (6, 7), (7, 6), (7, 4), (4, 7), (7, 7)}.

**(2)Transitive Closures:** Consider a relation R on a set A. The transitive closure R of a relation R of a relation R is the smallest transitive relation containing R.

Recall that R^{2} = Râ—¦ R and R^{n} = R^{n-1} â—¦ R. We define

The following Theorem applies:

**Theorem1:** R^{*} is the transitive closure of R

Suppose A is a finite set with n elements.

R^{*}= R âˆªR^{2}âˆª.....âˆª R^{n}

**Theorem 2:** Let R be a relation on a set A with n elements. Then

Transitive (R) = R âˆª R^{2}âˆª.....âˆª R^{n}

**Example1:** Consider the relation R = {(1, 2), (2, 3), (3, 3)} on A = {1, 2, 3}. Then

R^{2} = Râ—¦ R = {(1, 3), (2, 3), (3, 3)} and R^{3} = R^{2} â—¦ R = {(1, 3), (2, 3), (3, 3)} Accordingly,

Transitive (R) = {(1, 2), (2, 3), (3, 3), (1, 3)}

**Example2:** Let A = {4, 6, 8, 10} and R = {(4, 4), (4, 10), (6, 6), (6, 8), (8, 10)} is a relation on set A. Determine transitive closure of R.

**Solution:** The matrix of relation R is shown in fig:

Now, find the powers of M_{R} as in fig:

Hence, the transitive closure of M_{R} is M_{R*} as shown in Fig (where M_{R*} is the ORing of a power of M_{R}).

**Thus**, R^{*} = {(4, 4), (4, 10), (6, 8), (6, 6), (6, 10), (8, 10)}.

#### Note: While ORing the power of the matrix R, we can eliminate M_{Rn} because it is equal to M_{R*} if n is even and is equal to M_{R3} if n is odd.