Relation \( a\geq b \) on \( \mathbb{R} \)
📺 Video Explanation
📝 Question
Let relation \(S\) on the set of real numbers \(\mathbb{R}\) be defined by:
\[ aSb \iff a\geq b \]
Then, \(S\) is:
- A. an equivalence relation
- B. reflexive, transitive but not symmetric
- C. symmetric, transitive but not reflexive
- D. neither transitive nor reflexive but symmetric
✅ Solution
🔹 Reflexive Check
For every real number \(a\):
\[ a\geq a \]
✔ Reflexive.
🔹 Symmetric Check
If:
\[ a\geq b \]
symmetry would require:
\[ b\geq a \]
This is not always true.
Example:
\[ 5\geq3 \] but:
\[ 3\geq5 \] is false.
❌ Not symmetric.
🔹 Transitive Check
If:
\[ a\geq b \quad \text{and} \quad b\geq c \]
Then:
\[ a\geq c \]
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{Reflexive and transitive but not symmetric}} \]
✔ Correct option: B
🚀 Exam Shortcut
- Order relations like \( \geq \) are reflexive and transitive
- They are usually not symmetric
- So they are not equivalence relations