Largest Equivalence Relation on a Set
📺 Video Explanation
📝 Question
If \(R\) is the largest equivalence relation on a set \(A\) and \(S\) is any relation on \(A\), then:
- A. \(R \subset S\)
- B. \(S \subset R\)
- C. \(R=S\)
- D. none of these
✅ Solution
🔹 Largest Equivalence Relation
The largest equivalence relation on a set \(A\) is the universal relation:
\[ R=A\times A \]
because it contains every possible ordered pair.
🔹 Compare with Any Relation \(S\)
Any relation \(S\) on \(A\) is a subset of:
\[ A\times A \]
Since:
\[ R=A\times A \]
Therefore:
\[ S\subset R \]
🎯 Final Answer
\[ \boxed{S\subset R} \]
✔ Correct option: B
🚀 Exam Shortcut
- Largest equivalence relation = universal relation
- Universal relation contains all possible ordered pairs
- Every relation on set A is its subset