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On (A={1,2,3,4}), how many equivalence relations have exactly two equivalence classes?समुच्चय (A={1,2,3,4}) पर ऐसे समतुल्यता संबंधों की संख्या कितनी है जिनमें ठीक दो समतुल्यता वर्ग हों?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (7)(7)
Step 1: Exactly two classes means partitioning (4) elements into two non-empty disjoint parts. Step 2: There are (2^4-2=14) non-empty proper subsets, but each partition is counted twice. Step 3: Therefore the count is (\frac{14}{2}=7).चरण 1: ठीक दो वर्ग बनाने का अर्थ है (4) अवयवों को दो अरिक्त असंयुक्त भागों में बांटना। चरण 2: कुल अरिक्त उचित उपसमुच्चय (2^4-2=14) हैं, पर हर विभाजन दो बार गिना जाता है। चरण 3: इसलिए संख्या (\frac{14}{2}=7) होगी।
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