Question

Let \(R\) be the relation \(\{ (a,b)\mid a \ne b\} \) on the set of integers. What is the reflexive closure of \(R\)?

Short answer

The reflexive closure of \(R\) is \({\bf{Z}} \times {\bf{Z}}\).

Step-by-step solution

Given data

The \(R\) is \(\{ (a,b)\mid a \ne b\} \) and \(A = {\rm{ Set of integers }} = {\bf{Z}}\).

Concept of reflexive closure

The reflexive closure of\(R\)is the relation that contains all ordered pairs of\(R\)and to which all ordered pairs of the form\((a,a) \in R(a \in A)\)were added (when they were not present yet)\(R \cup \Delta = R \cup \{ (a,a)\mid a \in A\} \).

Find the reflexive closure of \(R\)

The reflexive closure of \(R\) is \(R \cup \Delta = R \cup \{ (a,a)\mid a \in A\} \)as follows:

\(\begin{array}{c}R \cup \Delta = \{ (a,b)\mid a \ne b\} \cup \{ (a,b)\mid a = b\} \\ = \{ (a,b)\mid a \ne b{\rm{ or }}a = b\} \\ = \{ (a,b)\mid a,b \in {\bf{Z}}\} \\ = {\bf{Z}} \times {\bf{Z}}\end{array}\)

The statement \(a \ne b\) or \(a = b{\rm{ }}\) is true for all pairs of integers.

Thus, the reflexive closure of \(R\) is \({\bf{Z}} \times {\bf{Z}}\).