If `R = {(a,b) : a+b=4}` is a relation on N, then R is

Options

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

Answer

To determine the properties of the relation \( R = \{(a, b) : a + b = 4\} \) on the set of natural numbers \( \mathbb{N} \), we will analyze whether it is reflexive, symmetric, anti-symmetric, and transitive.

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

Given the equation \( a + b = 4 \), we can find pairs \( (a, b) \) where both \( a \) and \( b \) are natural numbers (i.e., \( \mathbb{N} = \{1, 2, 3, \ldots\} \)).

- The possible pairs are:
- \( (1, 3) \) because \( 1 + 3 = 4 \)
- \( (2, 2) \) because \( 2 + 2 = 4 \)
- \( (3, 1) \) because \( 3 + 1 = 4 \)

Thus, the relation \( R \) can be expressed as:
\[ R = \{(1, 3), (2, 2), (3, 1)\} \]

### Step 2: Check for Reflexivity

A relation is reflexive if every element is related to itself, i.e., for all \( a \in \mathbb{N} \), \( (a, a) \in R \).

- Checking the pairs:
- \( (1, 1) \) is not in \( R \)
- \( (2, 2) \) is in \( R \)
- \( (3, 3) \) is not in \( R \)

Since not all elements are related to themselves, \( R \) is **not reflexive**.

### Step 3: Check for Symmetry

A relation is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).

- Checking the pairs:
- For \( (1, 3) \), \( (3, 1) \) is in \( R \)
- For \( (2, 2) \), \( (2, 2) \) is in \( R \)
- For \( (3, 1) \), \( (1, 3) \) is in \( R \)

Since all pairs satisfy the symmetric condition, \( R \) is **symmetric**.

### Step 4: Check for Anti-symmetry

A relation is anti-symmetric if whenever \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \).

- Checking the pairs:
- \( (1, 3) \) and \( (3, 1) \) are both in \( R \) but \( 1 \neq 3 \)

Since we found pairs where \( a \neq b \), \( R \) is **not anti-symmetric**.

### Step 5: Check for Transitivity

A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

- Checking the pairs:
- We have \( (1, 3) \) and \( (3, 1) \), but \( (1, 1) \) is not in \( R \)
- We also have \( (2, 2) \) which does not affect transitivity.

Since \( (1, 3) \) and \( (3, 1) \) do not lead to \( (1, 1) \) being in \( R \), \( R \) is **not transitive**.

### Conclusion

The relation \( R \) is:
- Not reflexive
- Symmetric
- Not anti-symmetric
- Not transitive

### Final Answer

Thus, the relation \( R \) is **symmetric**.

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