Discrete Mathematical Structures by Pearson

2-10 Discrete Mathematical Structures

2.2.7 Reflexive Closure

Let R be a given relation on a set A. The reflexive closure S of R is obtained by adding to R all

pairs of the form (a, a), which are not in R for every a ∈ A.

Example: Let R = {(1, 1), (2, 2), (1, 2), (2, 4)} be defined on the set A = {1, 2, 3, 4}. Find the

reflexive closure of R.

Solution: Reflexive closure S = R ∪ {(3, 3), (4, 4)} = {(1, 1), (2, 2), (1, 2), (2, 4), (3, 3), (4, 4)}

S is the reflexive closure of R.

2.2.8 Symmetric Closure

Let R be a given relation on the set A. The symmetric closure S of R is obtained by adding to R

all pairs of R

−1

, that is, all pairs (b, a) if (a, b) ∈ R, that is, S = R ∪ R

−1

.

Example: Let R = {(2, 2), (1, 2), (1, 3), (2, 4), (4, 4)} be a relation ...