Question

Find the error in the "proof" of the following "theorem."

"Theorem": Let \(R\) be a relation on a set \(A\) that is symmetric and transitive. Then \(R\) is reflexive.

"Proof": 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 \((b,a) \in R\).

Short answer

The error is "Take an element \(b \in {A^n}\)

Step-by-step solution

Given Data

Theorem: Let \(R\) be a relation on a set \(A\)that is symmetric and transitive. Then \(R\)is reflexive.

Proof: 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 \((b,a) \in R\).

Concept of the properties of relation

Reflexive: A relation \(R\) on a set \(A\) is called reflexive if \((a,a) \in R\) for every element \(a \in A\).

Symmetric: A relation \(R\) on a set \(A\) is called symmetric if \((b,a) \in R\) whenever \((a,b) \in R\), for all \(a,b \in A\).

Transitive: A relation \(R\) on a set \(A\) is called transitive if whenever \((a,b) \in R\) and \((b,c) \in R\), then \((a,c) \in R\), for all \(a,b,c \in A\).

Determine the error in the proof of the following theorem

\(R\)is a symmetric and transitive relation of a set \(A\).

The error in the proof is "Take an element \(b \in A\)", because it is possible that \(A\)is the empty set and thus it is possible that \(A\)does not contain any elements b.

Moreover, in exercise 8 we proved that if \(R = \emptyset \) and \(S = A\) nonempty, then \(R\)is symmetric, transitive and not reflexive.

Therefore, the error is "Take an element \(b \in {A^n}\).