Let `R` be the relation on the set R of all real numbers defined by a `R b` Iff `|a-b| le1.` Then `R` is

Options

A. reflexive and symmetric
B. symmetric only
C. transitive only
D. anti-symmetric only

Explanation

To determine the properties of the relation \( R \) defined on the set of all real numbers \( \mathbb{R} \) by \( a R b \) if and only if \( |a - b| \leq 1 \), we will check whether \( R \) is reflexive, symmetric, anti-symmetric, and transitive.

### Step 1: Check if \( R \) is Reflexive
A relation \( R \) is reflexive if every element is related to itself. This means we need to check if \( a R a \) holds true for all \( a \in \mathbb{R} \).

- For \( a R a \), we have:
\[ |a - a| \leq 1 \]
Simplifying this gives:
\[ |0| \leq 1 \]
This is true since \( 0 \leq 1 \).

Thus, \( R \) is reflexive.

### Step 2: Check if \( R \) is Symmetric
A relation \( R \) is symmetric if whenever \( a R b \) holds, then \( b R a \) also holds.

- Assume \( a R b \), which means:
\[ |a - b| \leq 1 \]
- We need to check if this implies \( |b - a| \leq 1 \):
\[ |b - a| = |a - b| \]
Since absolute value is symmetric, we have:
\[ |b - a| \leq 1 \]

Thus, \( R \) is symmetric.

### Step 3: Check if \( R \) is Anti-symmetric
A relation \( R \) is anti-symmetric if whenever \( a R b \) and \( b R a \) both hold, then \( a \) must equal \( b \).

- Assume \( a R b \) and \( b R a \):
\[ |a - b| \leq 1 \quad \text{and} \quad |b - a| \leq 1 \]
- This does not imply that \( a = b \). For example, let \( a = 1 \) and \( b = 2 \):
\[ |1 - 2| = 1 \quad \text{and} \quad |2 - 1| = 1 \]
Here, \( 1 \neq 2 \).

Thus, \( R \) is not anti-symmetric.

### Step 4: Check if \( R \) is Transitive
A relation \( R \) is transitive if whenever \( a R b \) and \( b R c \) hold, then \( a R c \) must also hold.

- Assume \( a R b \) and \( b R c \):
\[ |a - b| \leq 1 \quad \text{and} \quad |b - c| \leq 1 \]
- We need to check if this implies \( |a - c| \leq 1 \):
Using the triangle inequality:
\[ |a - c| \leq |a - b| + |b - c| \]
Thus:
\[ |a - c| \leq 1 + 1 = 2 \]
However, \( |a - c| \) being less than or equal to 2 does not guarantee that \( |a - c| \leq 1 \). For example, let \( a = 1 \), \( b = 2 \), and \( c = 3 \):
\[ |1 - 2| = 1 \quad \text{and} \quad |2 - 3| = 1 \]
But:
\[ |1 - 3| = 2 \quad \text{which is not} \leq 1. \]

Thus, \( R \) is not transitive.

### Conclusion
The relation \( R \) is:
- Reflexive: Yes
- Symmetric: Yes
- Anti-symmetric: No
- Transitive: No

Therefore, the relation \( R \) is reflexive and symmetric.