If `R={(a,b): |a+b|=a+b}` is a relation defined on a set `{-1, 0, 1}`, then R is

Options

A. reflexive
B. symmetric
C. anti symmetric
D. transitive

Answer

To determine the properties of the relation \( R = \{(a, b) : |a + b| = a + b\} \) defined on the set \( S = \{-1, 0, 1\} \), we will analyze whether \( R \) is reflexive, symmetric, anti-symmetric, or transitive.

### Step 1: Identify the relation \( R \)

We need to find all pairs \( (a, b) \) from the set \( S \) such that \( |a + b| = a + b \). This condition implies that \( a + b \) must be non-negative (since the absolute value of a number is equal to the number itself if the number is non-negative).

#### Possible pairs:
1. \( (0, 0) \): \( |0 + 0| = 0 \) (valid)
2. \( (1, 1) \): \( |1 + 1| = 2 \) (valid)
3. \( (-1, -1) \): \( |-1 + -1| = 2 \) (not valid)
4. \( (-1, 0) \): \( |-1 + 0| = 1 \) (not valid)
5. \( (-1, 1) \): \( |-1 + 1| = 0 \) (valid)
6. \( (0, 1) \): \( |0 + 1| = 1 \) (valid)
7. \( (1, 0) \): \( |1 + 0| = 1 \) (valid)

From this analysis, we find the valid pairs:
- \( (0, 0) \)
- \( (1, 1) \)
- \( (-1, 1) \)
- \( (0, 1) \)
- \( (1, 0) \)

Thus, the relation \( R \) can be expressed as:
\[ R = \{(0, 0), (1, 1), (-1, 1), (0, 1), (1, 0)\} \]

### Step 2: Check for reflexivity

A relation is reflexive if every element is related to itself. This means that for all \( x \in S \), \( (x, x) \) must be in \( R \).

- \( (0, 0) \in R \) (valid)
- \( (1, 1) \in R \) (valid)
- \( (-1, -1) \notin R \) (not valid)

Since \( (-1, -1) \) is not in \( R \), the relation is **not reflexive**.

### Step 3: Check for symmetry

A relation is symmetric if for every \( (x, y) \in R \), \( (y, x) \) must also be in \( R \).

- \( (0, 0) \) is symmetric with itself.
- \( (1, 1) \) is symmetric with itself.
- \( (-1, 1) \) implies \( (1, -1) \) (not in \( R \)).
- \( (0, 1) \) implies \( (1, 0) \) (in \( R \)).
- \( (1, 0) \) implies \( (0, 1) \) (in \( R \)).

Since \( (-1, 1) \) does not have \( (1, -1) \) in \( R \), the relation is **not symmetric**.

### Step 4: Check for anti-symmetry

A relation is anti-symmetric if for every \( (x, y) \in R \) and \( (y, x) \in R \), it must hold that \( x = y \).

- From our pairs, \( (0, 1) \) and \( (1, 0) \) are in \( R \) but \( 0 \neq 1 \).

Thus, the relation is **not anti-symmetric**.

### Step 5: Check for transitivity

A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \).

- Consider \( (0, 1) \) and \( (1, 0) \) in \( R \). We have \( (0, 0) \) in \( R \), which is valid.
- However, consider \( (-1, 1) \) and \( (1, 0) \). We have \( (-1, 0) \) which is not in \( R \).

Thus, the relation is **not transitive**.

### Conclusion

The relation \( R \) defined on the set \( S = \{-1, 0, 1\} \) is:
- Not reflexive
- Not symmetric
- Not anti-symmetric
- Not transitive

The final answer is that \( R \) is **not reflexive, symmetric, anti-symmetric, or transitive**.