Let S = Set of all women in the world. Define R as follows:
`a,b in S, aRb` if a is mother of b. Then R is

Options

A. reflexive
B. symmetric
C. not a relation
D. none of these

Answer

To determine the properties of the relation \( R \) defined on the set \( S \) (the set of all women in the world), where \( aRb \) means "a is the mother of b", we will analyze the relation in terms of reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \) in the set \( S \), the relation \( aRa \) holds. This means that every woman must be her own mother.

- In our case, it is impossible for a woman to be her own mother. Therefore, \( aRa \) does not hold for any woman \( a \).

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

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair of elements \( a \) and \( b \) in the set \( S \), whenever \( aRb \) holds, then \( bRa \) must also hold. This means if \( a \) is the mother of \( b \), then \( b \) must also be the mother of \( a \).

- In our case, if \( a \) is the mother of \( b \), it does not imply that \( b \) is the mother of \( a \). For example, if \( a \) is a mother and \( b \) is her child, then \( b \) cannot be the mother of \( a \).

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

### Step 3: Check for Transitivity
A relation \( R \) is transitive if for any elements \( a, b, c \) in the set \( S \), whenever \( aRb \) and \( bRc \) hold, then \( aRc \) must also hold. This means if \( a \) is the mother of \( b \) and \( b \) is the mother of \( c \), then \( a \) must be the mother of \( c \).

- In our case, if \( a \) is the mother of \( b \) and \( b \) is the mother of \( c \), then \( a \) is indeed the grandmother of \( c \), but not the mother. Therefore, \( aRc \) does not hold.

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

### Final Conclusion
The relation \( R \) defined as "a is the mother of b" is neither reflexive, nor symmetric, nor transitive.

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