Question

To prove the error in the given proof a theorem.

Short answer

The error in given proof a theorem is "Take an element \(b \in A\) such that \((a,b) \in R\) ".

Step-by-step solution

 Given

Let \(a \in A\). Take an element \(b \in A\) such that \((a,b) \in R\).

The Concept of reflexive relation

Let the relation\(R\)on set\(A\)that is symmetric and transitive. Then\(R\)is reflexive.

Determine the relation

Let the relation\(R\)on set\(A\)that is symmetric and transitive. Then\(R\)is reflexive

let\(a \in A\). Take an element\(b \in A\)such that\((a,b) \in R\). because\(R\)is symmetric, we also have\((b,a) \in R\). Now using the transitive property, we can conclude that\((a,a) \in R\). because\((a,b) \in R\)and\((b,a) \in R\).

Consider the statement "relation\(R\)on set\(A\)which is symmetric and transitive. Then\(R\)is reflexive"

If we observe the proof, if the element\(b\)is not exist, then the proof is correct.

Take an element \(b \in A\) such that \((a,b) \in R\). is wrong. Because \(R\) is reflexive.