Let S be the set of all points in a plane. Let R be a relation on S such that for any two points a and b, a R b iff b is within 1 centimetre from a. Check R for reflexivity, symmetry and transitivity.

Answer

To analyze the relation \( R \) on the set \( S \) of all points in a plane, we need to check if it satisfies the properties of reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element is related to itself. In our case, we need to check if for every point \( A \) in \( S \):

\[ A R A \]

This means we need to check if the distance between point \( A \) and itself is less than 1 cm. The distance between any point and itself is 0 cm, which is indeed less than 1 cm. Therefore, we can conclude that:

\[ A R A \text{ is true for all } A \in S \]

Thus, the relation \( R \) is reflexive.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for any two points \( A \) and \( B \) in \( S \):

\[ A R B \implies B R A \]

This means if \( A \) is related to \( B \) (i.e., the distance between \( A \) and \( B \) is less than 1 cm), we need to check if \( B \) is also related to \( A \). The distance between \( A \) and \( B \) is the same as the distance between \( B \) and \( A \). Therefore, if:

\[ \text{Distance}(A, B) < 1 \text{ cm} \]

It follows that:

\[ \text{Distance}(B, A) < 1 \text{ cm} \]

Thus, \( B R A \) holds true. Therefore, the relation \( R \) is symmetric.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if for any three points \( A \), \( B \), and \( C \) in \( S \):

\[ (A R B \text{ and } B R C) \implies A R C \]

This means if \( A \) is related to \( B \) and \( B \) is related to \( C \), we need to check if \( A \) is related to \( C \).

Assuming:

1. \( A R B \) implies \( \text{Distance}(A, B) < 1 \) cm
2. \( B R C \) implies \( \text{Distance}(B, C) < 1 \) cm

Using the triangle inequality, we can say:

\[ \text{Distance}(A, C) \leq \text{Distance}(A, B) + \text{Distance}(B, C) \]

Thus, we have:

\[ \text{Distance}(A, C) < 1 + 1 = 2 \text{ cm} \]

However, for \( A R C \) to hold, we need \( \text{Distance}(A, C) < 1 \) cm, which is not guaranteed. Therefore, \( R \) is not transitive.

### Conclusion
- The relation \( R \) is **reflexive**.
- The relation \( R \) is **symmetric**.
- The relation \( R \) is **not transitive**.