Let a relation `R_1` on the set `R` of real numbers be defined as `(a ,\ b) in R_1<=>1+a b >0` for all `a ,\ b in R` . Show that `R_1` is reflexive and symmetric but not transitive.

Explanation

To show that the relation \( R_1 \) defined on the set of real numbers \( R \) by \( (a, b) \in R_1 \iff 1 + ab > 0 \) is reflexive, symmetric, but not transitive, we will go through each property step by step.

### Step 1: Check if \( R_1 \) is Reflexive
A relation is reflexive if for every element \( a \in R \), \( (a, a) \in R_1 \).

**Proof:**
- For \( (a, a) \), we need to check if \( 1 + a \cdot a > 0 \).
- This simplifies to \( 1 + a^2 > 0 \).
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