Question

To prove that \(R\) is reflexive if and only if \({R^{ - 1}}\) is reflexive.

Short answer

The relation \(R\) is reflexive if and only if \({R^{ - 1}}\) is reflexiveis proved.

Step-by-step solution

 Given

The proof is given in two pairs

(1)If\(R\)is reflexive then\({R^{ - 1}}\)is reflexive.

(2) If \({R^{ - 1}}\) is reflexive then \(R\) is reflexive.

The Concept ofreflexive relation

A homogeneous binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself.

Determine the relation

To prove (1):

Let\(R\)be a relation on set\(A\)

Suppose\(R\)is reflexive

\( \Rightarrow (a,a) \in R\), for all\(a \in A\)

Since\({R^{ - 1}}\)is inverse relation to\(R\)

\( \Rightarrow (a,a) \in {R^{ - 1}}\), for all\(a \in A\)

Hence\({R^{ - 1}}\)is reflexive is proved.

To prove (2): The Converse

Suppose\({R^{ - 1}}\)is reflexive

\( \Rightarrow (a,a) \in {R^{ - 1}}\), for all\(a \in A\)

Since\({R^{ - 1}}\)is inverse relation to\(R\)

\( \Rightarrow (a,a) \in R\), for all\(a \in A\)

Thus,\(R\)is reflexive.

Hence\(R\)is reflexive is proved.

Therefore, \(R\) is reflexive if and only \({R^{ - 1}}\) is reflexive\(R = {R^{ - 1}}\) if and only if \(R\) is symmetric.