Show that the relation `R` on the set `A={1,\ 2,\ 3}` given by `R={(1,\ 2),\ (2,\ 1)}` is symmetric but neither reflexive nor transitive.

Explanation

To determine the properties of the relation \( R \) on the set \( A = \{1, 2, 3\} \) given by \( R = \{(1, 2), (2, 1)\} \), we will check if the relation is reflexive, symmetric, and transitive.

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element \( a \in A \) is related to itself, meaning \( (a, a) \in R \) for all \( a \).

- For \( a = 1 \): \( (1, 1) \notin R \)
- For \( a = 2 \): \( (2, 2) \notin R \)
- For \( a = 3 \): \( (3, 3) \notin R \)
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