If R is a relation on R (set of all real numbers) defined by a R b iff `ageb`, then R is

Options

A. an equivalence relation
B. reflexive, transitive but not symmetric
C. symmetric, transitive nut not reflexive
D. neither reflexive nor transitive but symmetric

Answer

To determine the properties of the relation \( R \) defined on the set of real numbers \( \mathbb{R} \) by \( a R b \) if and only if \( a \geq b \), we will analyze the relation with respect to three properties: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Reflexivity**:
- A relation \( R \) is reflexive if for every element \( a \in \mathbb{R} \), \( a R a \) holds true.
- Here, \( a R a \) translates to \( a \geq a \), which is always true for any real number \( a \).
- Therefore, the relation \( R \) is reflexive.

2. **Symmetry**:
- A relation \( R \) is symmetric if for every \( a, b \in \mathbb{R} \), whenever \( a R b \) holds, then \( b R a \) must also hold.
- In our case, if \( a R b \) means \( a \geq b \), it does not imply that \( b \geq a \) unless \( a = b \).
- For example, if \( a = 3 \) and \( b = 2 \), then \( 3 \geq 2 \) is true, but \( 2 \geq 3 \) is false.
- Hence, the relation \( R \) is not symmetric.

3. **Transitivity**:
- A relation \( R \) is transitive if for every \( a, b, c \in \mathbb{R} \), whenever \( a R b \) and \( b R c \) hold, then \( a R c \) must also hold.
- Here, if \( a R b \) means \( a \geq b \) and \( b R c \) means \( b \geq c \), then it follows that \( a \geq b \) and \( b \geq c \) implies \( a \geq c \).
- Therefore, the relation \( R \) is transitive.

### Conclusion:
- The relation \( R \) is reflexive and transitive, but not symmetric.

### Final Answer:
The relation \( R \) is reflexive and transitive, but not symmetric.

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