Question:

Step 1: Check if the relation is reflexive. A relation $R$ is reflexive if $aRa$ for all $a \in S$. We need to verify that $|a - a| \leq 1$. $$|a - a| = 0 \leq 1$$ Since this is true for all real numbers $a$, the relation $R$ is **reflexive**. ✓ Step 2: Check if the relation is symmetric. A relation $R$ is symmetric if whenever $aRb$, then $bRa$. We need to verify that if $|a - b| \leq 1$, then $|b - a| \leq 1$. By the property of absolute values, we know that: $$|a - b| = |b - a|$$ Therefore, if $|a - b| \leq 1$, then automatically $|b - a| \leq 1$. The relation $R$ is **symmetric**. ✓ Step 3: Check if the relation is transitive. A relation $R$ is transitive if whenever $aRb$ and $bRc$, then $aRc$. We need to check if $|a - b| \leq 1$ and $|b - c| \leq 1$ together imply $|a - c| \leq 1$. Let us test with a counterexample: Let $a = 1$, $b = 2$, and $c = 3$. - Check $aRb$: $|1 - 2| = 1 \leq 1$ ✓ - Check $bRc$: $|2 - 3| = 1 \leq 1$ ✓ - Check $aRc$: $|1 - 3| = 2 \not\leq 1$ ✗ Since we have $aRb$ and $bRc$, but $aRc$ does not hold, the relation $R$ is **not transitive**. ✗ Step 4: Conclude the answer. The relation $R$ is reflexive and symmetric, but not transitive. **The answer is Option 3: reflexive and symmetric but not transitive.**