Let S be a non-empty set of children in a family and R be a relation on S defined by a R b iff a is a brother of b then R is

Options

A. symmetric but not transitive
B. not symmetric but transitive
C. neither symmelric nor fransitive
D. both symmetric and transitive

Answer

To determine the properties of the relation \( R \) defined on the set \( S \) of children in a family, where \( a R b \) if \( a \) is a brother of \( b \), we will analyze the relation for reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element is related to itself. In this case, for \( R \) to be reflexive, every child \( a \) in the set \( S \) must be a brother of themselves.

- **Analysis**: A child cannot be their own brother. Therefore, \( a R a \) is not true for any \( a \) in \( S \).

**Conclusion**: The relation \( R \) is **not reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if whenever \( a R b \) holds, then \( b R a \) must also hold.

- **Analysis**: If \( a \) is a brother of \( b \) (i.e., \( a R b \)), it does not necessarily mean that \( b \) is a brother of \( a \). For example, if \( a \) is a male child and \( b \) is a female child, then \( b \) cannot be a brother of \( a \).

**Conclusion**: The relation \( R \) is **not symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \) hold, then \( a R c \) must also hold.

- **Analysis**: Suppose \( a R b \) (i.e., \( a \) is a brother of \( b \)) and \( b R c \) (i.e., \( b \) is a brother of \( c \)). In this case, \( a \) and \( b \) are brothers, and \( b \) and \( c \) are brothers. This implies that \( a \) and \( c \) could also be brothers, making \( a R c \) true.

**Conclusion**: The relation \( R \) is **transitive**.

### Final Conclusion
The relation \( R \) defined by "a is a brother of b" is:
- Not reflexive
- Not symmetric
- Transitive

Thus, the relation \( R \) is **transitive**.

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