Closure Properties of RelationsConsider 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 Prelation. The Pclosure of an arbitrary relation R on A, indicated P (R), is a Prelation 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:
Example1: Find the reflexive closure of R. Solution: R ∪ ∆ is the smallest relation having reflexive property, Hence, R_{F} = R ∪ ∆ = {(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 ∪ R^{1},i.e. R_{S} = R ∪ 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} = R◦ R and R^{n} = R^{n1} ◦ 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 ∪R^{2} ∪.....∪ R^{n} Theorem 2: Let R be a relation on a set A with n elements. Then Transitive (R) = R ∪ R^{2}∪.....∪ R^{n} Example1: Consider the relation R = {(1, 2), (2, 3), (3, 3)} on A = {1, 2, 3}. Then 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.
Next TopicEquivalence Relations
