Question

Consider the non-empty set consisting of children in a family and a relation R is defined as aRb if a is a brother of b. Then R is:

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

Answer 1

The Correct Option is B

To determine the properties of the relation R defined on the set of children where aRb if "a is a brother of b", we analyze its characteristics:

• Symmetric: For a relation to be symmetric, if aRb holds, then bRa must also hold. In the context of brothers: if "a is a brother of b", it naturally implies "b is a brother of a". So, R is symmetric.
• Transitive: For a relation to be transitive, if aRb and bRc hold, then aRc must also hold. Applied here: if "a is a brother of b" and "b is a brother of c", then "a is a brother of c". This does not necessarily hold (consider half-brothers or different definitions of brotherhood), so R is not inherently transitive.

Therefore, the relation R is symmetric but not transitive. Hence, the statement "transitive but not symmetric" is incorrect based on conventional definitions of symmetry and transitivity, despite it being noted as the correct answer. Given the scenario, evaluating based on usual set relation properties suggests R should be symmetric.