The relation 'is not equal to' is defined on R, is

Options

A. reflexive only
B. symmetric only
C. transitive only
D. equivalence.

Answer

To determine the properties of the relation "is not equal to" defined on the set of real numbers \( R \), we will check if this relation is reflexive, symmetric, and transitive.

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

For the relation "is not equal to":
- We need to check if \( a \neq a \) for any real number \( a \).
- This is not true since \( a \) is always equal to itself.

**Conclusion**: The relation "is not equal to" is **not reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair \( (a, b) \) in \( R \), the pair \( (b, a) \) is also in \( R \).

For the relation "is not equal to":
- If \( a \neq b \), then it is also true that \( b \neq a \).
- Therefore, if \( (a, b) \) is in the relation, then \( (b, a) \) is also in the relation.

**Conclusion**: The relation "is not equal to" is **symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).

For the relation "is not equal to":
- Suppose \( a \neq b \) and \( b \neq c \).
- We cannot conclude that \( a \neq c \). For example, let \( a = 1 \), \( b = 2 \), and \( c = 1 \). Here, \( 1 \neq 2 \) and \( 2 \neq 1 \), but \( 1 = 1 \).

**Conclusion**: The relation "is not equal to" is **not transitive**.

### Final Conclusion
Based on the evaluations:
- The relation "is not equal to" is **not reflexive**.
- The relation "is not equal to" is **symmetric**.
- The relation "is not equal to" is **not transitive**.

Thus, the correct answer is that the relation "is not equal to" is **symmetric**.

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