Question

Consider the non-empty set consisting of children in a house. Consider a relation \(R\): \(xRy\) if \(x\) is brother of \(y\). The relation \(R\) is:

• A. Symmetric but not transitive
• B. Transitive but not symmetric and reflexive
• C. Neither symmetric nor transitive
• D. Both symmetric and transitive

Hint:

Common properties of relations:
\textbf{Symmetric}: relation works both ways
\textbf{Transitive}: relation passes through an intermediate element
Family relations like \emph{brother of} are usually neither symmetric nor transitive

Answer 1

The Correct Option is C

Step 1: Check symmetry A relation is symmetric if \(xRy \Rightarrow yRx\). If \(x\) is brother of \(y\), then \(y\) need not be brother of \(x\) (since \(y\) may be a sister). \[ \Rightarrow \text{Relation is not symmetric} \]
Step 2: Check transitivity A relation is transitive if \(xRy\) and \(yRz \Rightarrow xRz\). If \(x\) is brother of \(y\) and \(y\) is brother of \(z\), it does not necessarily imply that \(x\) is brother of \(z\) in all cases (family relations may differ). \[ \Rightarrow \text{Relation is not transitive} \]
Step 3: Hence, the relation is neither symmetric nor transitive.