Let S be the set of all points in a plane and R be a relation on S defines as `R={(P,Q):` distance between `P and Q` is less than 2 units} Show that `R` is reflexive and symmetric but not transitive.

Explanation

To show that the relation \( R \) defined on the set \( S \) of all points in a plane, where \( R = \{(P, Q) : \text{distance between } P \text{ and } Q < 2 \text{ units}\} \), is reflexive, symmetric, but not transitive, we will go through each property step by step.

### Step 1: Proving Reflexivity

**Definition of Reflexivity:** A relation \( R \) is reflexive if for every element \( P \) in set \( S \), the pair \( (P, P) \) is in \( R \).

**Proof:**
- Let \( P \) be any point in the plane \( S \).
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